Analyses of Sandwich Beams and Plates with Viscoelastic Cores [PhD Thesis ed.]

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ABSTRACT Title of Dissertation: Analyses of Sandwich Beams and Plates

with Viscoelastic Cores Gang Wang, Doctor of Philosophy, 2001 Dissertation directed by: Associate Professor Norman M. Wereley Department of Aerospace Engineering

A hybrid damping scheme using passive constrained damping layers (PCLD), and surface bonded piezoceramic actuators was proposed for interior cabin noise and vibration control in helicopters. In order to evaluate the performance of these treatments, we need to understand the dynamic behavior of sandwich structures. The analyses of sandwich structures are complicated by the frequency dependent stiffness and damping properties of viscoelastic materials. The methods developed in this thesis specifically deal with finite element methods and assumed modes methods to this problem. A spectral finite element method (SFEM) was developed in the frequency domain for sandwich beam analysis. The results of natural frequencies and frequency responses for two cantilevered beams with different span of PCLD treatments were presented and validated by experimental results and other analyses;

including the assumed modes method (AM), and conventional finite element method (CFEM). The SFEM method implicitly accounts for frequency dependent stiffness and damping of viscoelastic materials. However, CFEM and AM method have to use additional internal dissipation coordinates to account for these properties. The Golla-Hughes-McTavish (GHM) damping method was used in both analyses. Also SFEM improves accuracy of frequency predictions compared to the results of CFEM and AM method because of its higher order interpolation functions. We expected to extend SFEM method to two-dimensional sandwich plate structures. But it is extremely difficult to solve the governing equations for a sandwich plate. An alternative method was developed to update the traditional AM method by using plate mode shapes. The plate mode shape functions were solved directly based on the Kantorovich variational method for both transverse bending and in-plane vibration of isotropic rectangular plates. These plate mode shapes were employed to calculate sandwich plates in AM method. The results of natural frequencies, loss factors and frequency response functions were calculated and validated by experimental data and the results by using beam and rod mode shapes. The comparable results were achieved for both analyses with less modes in the case of using plate mode shapes.

Analyses of Sandwich Beams and Plates with Viscoelastic Cores by Gang Wang

Dissertation submitted to the Faculty of the Graduate School of The University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2001

Thesis Committee: Associate Professor Norman M. Wereley, Chairman/Advisor Professor Amr Baz Professor Inderjit Chopra Professor Sung W. Lee Associate Professor Darryll J. Pines

c Copyright by
Gang Wang 2001

DEDICATION

To my parents and my teachers.

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ACKNOWLEDGEMENTS

My sincere gratitude and appreciation goes to my advisor, Dr. Norman Wereley. His kindness, encouragement, and support, helped my walk through this path. His personal concerns for my family are especially appreciated. I would also like to thank my dissertation committee members, Dr. Baz, Dr. Chopra, Dr. Lee. and Dr. Pines, for their suggestions, and their enthusiasm in my research. Over the course of my graduate studies, I have shared happy moments and tough times with my colleagues at the Alfred Gessow Rotorcraft Center. I am grateful for this. I wish to thank my wife, Ying, for her patience, help, and love during those years. Many thanks to Dr. Chang, for his helpful discussions on mathematics in my research as well as his personal concerns for my family.

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Above all, I give thanks to God. Finally, this research was supported by U.S Army Research Office under the FY96 MURI in Active Control of Rotorcraft Vibration and Acoustics, with Dr. Gary Anderson and Dr. Tom Doligalski serving as technical monitors. Lab equipment support was provided under the FY96 Defense University Research Instrumentation Program (DURIP) Contract No. DAAH-0496-10301, and Dr. Gary Anderson serving as technical monitor.

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TABLE OF CONTENTS

LIST OF TABLES

ix

LIST OF FIGURES

xiv

1 Introduction

1

1.1

Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . .

1

1.2

State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2.1

Viscoelastic Materials . . . . . . . . . . . . . . . . . . . . .

4

1.2.2

Sandwich Beams . . . . . . . . . . . . . . . . . . . . . . .

7

1.2.3

Sandwich Plates . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3

1.4

Scope of the Present Research . . . . . . . . . . . . . . . . . . . . 12 1.3.1

Sandwich Beam . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.2

Sandwich Plate . . . . . . . . . . . . . . . . . . . . . . . . 15

Organization

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Viscoelastic Materials 2.1

19

Characteristics of Viscoelastic Materials . . . . . . . . . . . . . . 20

v

2.2

Classical Damping Models . . . . . . . . . . . . . . . . . . . . . . 22

2.3

Modern Damping Models . . . . . . . . . . . . . . . . . . . . . . . 25

2.4

2.3.1

Fractional Derivatives Model . . . . . . . . . . . . . . . . . 26

2.3.2

AFT and ADF Models . . . . . . . . . . . . . . . . . . . . 27

2.3.3

Golla-Hughes-McTavish Model . . . . . . . . . . . . . . . . 28

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Comparison of Analyses of Sandwich Beams 3.1

3.2

40

Assumptions and Governing Equations . . . . . . . . . . . . . . . 41 3.1.1

Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.2

Governing Equations . . . . . . . . . . . . . . . . . . . . . 43

Spectral Finite Element Method . . . . . . . . . . . . . . . . . . . 45 3.2.1

Isotropic Rod and Beam . . . . . . . . . . . . . . . . . . . 47

3.2.2

Sandwich Beam . . . . . . . . . . . . . . . . . . . . . . . . 50

3.3

Conventional Finite Element Method . . . . . . . . . . . . . . . . 57

3.4

Assumed Modes Method . . . . . . . . . . . . . . . . . . . . . . . 59

3.5

Solution Type/Methods . . . . . . . . . . . . . . . . . . . . . . . 61

3.6

Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.7

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.7.1

Modal Frequency Predictions . . . . . . . . . . . . . . . . 64

3.7.2

Number of Elements . . . . . . . . . . . . . . . . . . . . . 65

3.7.3

Frequency Response Functions . . . . . . . . . . . . . . . . 66

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3.8

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Analyses of Sandwich Plates: Part I 4.1

4.2

4.3

Assumptions and Governing Equations . . . . . . . . . . . . . . . 81 4.1.1

Asssumptions . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.2

Sandwich Plate Energies and Governing Equations . . . . 83

Assumed Modes Method Using Beam and Rod modes . . . . . . . 88 4.2.1

Analytical Validation: Simply Supported . . . . . . . . . . 90

4.2.2

Experimental Validation: All Four Sides Clamped . . . . . 92

4.2.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5 Plate Mode Shapes 5.1

80

110

Plate In-plane Mode Shape . . . . . . . . . . . . . . . . . . . . . . 112 5.1.1

Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.1.2

Validation and Results . . . . . . . . . . . . . . . . . . . . 122

5.1.3

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.2

Plate Bending Vibration . . . . . . . . . . . . . . . . . . . . . . . 133

5.3

Results for Plate Bending and In-plane Mode Shape Functions . . 139

6 Analyses of Sandwich Plate: Part II

146

6.1

Experimental Set-up . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

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6.3

6.2.1

Aluminum Plate . . . . . . . . . . . . . . . . . . . . . . . 153

6.2.2

Plate with PCLD Treatment . . . . . . . . . . . . . . . . . 155

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

7 Summary and Conclusions

175

7.1

Sandwich Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

7.2

Sandwich Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

7.3

Recommendations for Future Research . . . . . . . . . . . . . . . 178

A Mass and Stiffness Matrices

180

A.1 Mass Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 A.2 Stiffness Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Bibliography

187

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LIST OF TABLES

3.1

Beam and actuator constants . . . . . . . . . . . . . . . . . . . . 71

3.2

Comparison of predicted and measured modal frequencies of the beam specimen 1 with a 75% PCLD treatment. N is the number of finite elements; Nb is the number of bending modes and Ne is the number of extension modes used in AM method. . . . . . . . 71

3.3

Comparison of predicted and measured modal frequencies for beam specimen 2 with a 50% PCLD treatment. N is the number of finite elements; Nb is the number of bending modes and Ne is the number of extension modes used in AM method. . . . . . . . . . . 72

4.1

Material constants for a simply-supported sandwich plate . . . . . 92

4.2

Mode number mapping table . . . . . . . . . . . . . . . . . . . . . 100

4.3

Validation of sandwich model against theoretical solution . . . . . 101

4.4

Curve fitting of mini-oscillator parameters used in GHM method for the viscoelastic materials 3M ISD112 at different temperatures 102

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4.5

Calibration of experimental set-up: the influence of plate thickness on accuracy of the experiments. Aluminum plate dimensions: 67.31 cm x 52.07 cm x t (26.5” x 20.5” x t) . . . . . . . . . . . . . 103

4.6

Experimental validation using symmetric clamped sandwich II . . 104

4.7

Experimental validation for 67.31 cm x 52.07 cm x (0.04cm - VEM - 0.08cm) (26.5” x 20.5” x (1/64” Al - VEM - 1/32” Al)) asymmetric clamped sandwich plate; nb = 25, ne = 25, at 20◦ . . . . . . 105

4.8

Experimental validation for 67.31 cm x 52.07 cm x (0.04cm - VEM - 0.08cm) (26.5” x 20.5” x (1/64” Al - VEM - 1/32” Al)) asymmetric clamped sandwich plate; nb = 25, ne = 25, at 20◦ . . . . . . 106

4.9

Effect of the number of assumed modes on the modal predictions for the symmetric sandwich plate, at 20◦ . . . . . . . . . . . . . . . 107

5.1

Admissible rod mode shape functions . . . . . . . . . . . . . . . . 125

5.2

Natural frequencies of in-plane vibration of a rectangular plate with CCCC boundary conditions . . . . . . . . . . . . . . . . . . 126

5.3

Natural frequencies of in-plane vibration of a rectangular plate with CCCF boundary conditions . . . . . . . . . . . . . . . . . . 127

5.4

Natural frequencies of in-plane vibration of a rectangular plate with CFCF boundary conditions; Modal number 0 denotes the ”rigid” mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

x

5.5

The parameters in mode shape functions of a rectangular plate bending vibration under CFCF boundary condition I: where Wmn (x, y) = Xwm Ywn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.6

The parameters in mode shape functions of a rectangular plate bending vibration under CFCF boundary condition II: where Wmn (x, y) = Xwm Ywn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

5.7

The parameters in mode shape functions of a rectangular plate inplane vibration under CFCF boundary condition I: where Umn (x, y) = Xum Yun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.8

The parameters in mode shape functions of a rectangular plate inplane vibration under CFCF boundary condition II: where Umn (x, y) = Xum Yun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.9

The parameters in mode shape functions of a rectangular plate in-plane vibration under CFCF boundary condition III: where Vmn (x, y) = Xvm Yvn . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.10 The parameters in mode shape functions of a rectangular plate in-plane vibration under CFCF boundary condition IV: where Vmn (x, y) = Xvm Yvn . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.1

Coordinates of the 15 measured locations for an aluminum plate under CFCF boundary conditions; x and y are in inches . . . . . 152

xi

6.2

Coordinates of the 15 measured locations for a plate with PCLD treatment under CFCF boundary conditions; x and y are in inches 152

6.3

Bending frequency results for an aluminum plate with CFCF boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.4

Experimental results of bending mode shape functions for an aluminum plate with CFCF boundary conditions, 15 tested locations from mode 1 to 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

6.5

Experimental results of bending mode shape functions for an aluminum plate with CFCF boundary conditions, 15 tested locations from mode 5 to 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.6

Bending frequency results for a plate with PCLD treatment, as shown in Figure 6.3; in analysis I, 25 modes for each displacement were assumed and it leads to 500 degrees of freedom; in analysis II, 16 modes for each displacement were used for a total of 320 degrees of freedom. . . . . . . . . . . . . . . . . . . . . . . . . . . 165

6.7

Loss factor results for a plate with PCLD treatment, as shown in Figure 6.3; in analysis I, 25 modes for each displacement were assumed and it leads to 500 degrees of freedom; in analysis II, 16 modes for each displacement were used for a total of 320 degrees of freedom.

6.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Experimental results of bending mode shape functions for a plate with PCLD treatment; 15 tested locations from mode 1 to 3 . . . 167

xii

6.9

Experimental results of bending mode shape functions for the plate with PCLD treatment; 15 tested locations from mode 4 to 5 168

xiii

LIST OF FIGURES

2.1

Nomogram of the viscoelastic material, 3M ISD 112. . . . . . . . . 36

2.2

Storage Modulus and Loss Factor Vs. frequency at temperature 20o C for the viscoelastic material 3M ISD 112. . . . . . . . . . . . 36

2.3

Classical models of viscoelastic materials . . . . . . . . . . . . . . 37

2.4

Creep functions for three models . . . . . . . . . . . . . . . . . . . 37

2.5

Relaxation functions for three models . . . . . . . . . . . . . . . . 38

2.6

The mini-oscillators mechanical analogy in GHM method . . . . . 38

2.7

The GHM prediction of complex shear modulus using three minioscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1

Specimen 1: the PCLD treatment covers 75% of the total length of the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.2

Specimen 2: the PCLD treatment covers 50% of the total length of the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3

Cross section of beam with PCLD treatment . . . . . . . . . . . . 74

3.4

Deflection of beam with PCLD treatment . . . . . . . . . . . . . . 74

xiv

3.5

Nodal degrees of freedom in SFEM . . . . . . . . . . . . . . . . . 74

3.6

Nodal degrees of freedom in CFEM . . . . . . . . . . . . . . . . . 75

3.7

Experimental set up for beam with PCLD treatments . . . . . . . 75

3.8

Number of elements used in SFEM and CFEM for 50% PCLD beam 76

3.9

The effects of number of elements on modal frequencies for specimen 1 having 75% PCLD treatment . . . . . . . . . . . . . . . . 76

3.10 The effects of number of elements on modal frequencies for specimen 2 having 50% PCLD treatment . . . . . . . . . . . . . . . . 77 3.11 Frequency Response function from the piezoelectric voltage input to the tip displacement output: the PCLD treatment covers 75% of the length of the base beam. . . . . . . . . . . . . . . . . . . . 78 3.12 Frequency Response function from the piezoelectric voltage input to the tip displacement output: the PCLD treatment covers 50% of the length of the base beam. . . . . . . . . . . . . . . . . . . . 79 4.1

Sandwich plate and layer displacements . . . . . . . . . . . . . . . 82

4.2

Experimental setup for plate test . . . . . . . . . . . . . . . . . . 94

4.3

The temperature effects on the frequencies and system loss factors for a symmetric clamped sandwich plate; ne = 25, nb = 25. . . . . 108

4.4

The temperature effects on the frequencies and system loss factors for the first asymmetric clamped sandwich plate; ne = 25, nb = 25. 108

xv

4.5

The temperature effects on the frequencies and system loss factors for the second asymmetric clamped sandwich plate; ne = 25, nb = 25. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.1

Schematic of rectangular plate under in-plane vibration . . . . . . 113

5.2

Three configurations of rectangular plate under in-plane vibration 129

5.3

Mode shapes of in-plane vibration of a rectangular plate with CCCC boundary conditions . . . . . . . . . . . . . . . . . . . . . 130

5.4

Mode shapes of in-plane vibration of a rectangular plate with CCCF boundary conditions . . . . . . . . . . . . . . . . . . . . . 131

5.5

Mode shapes of in-plane vibration of a rectangular plate with CFCF boundary conditions . . . . . . . . . . . . . . . . . . . . . 132

5.6

Schematic of rectangular plate bending vibration

. . . . . . . . . 133

5.7

A uniform rectangular plate with CFCF boundary conditions . . . 139

6.1

Schematic of plate testing set-up

6.2

Diagram of clamping fixture . . . . . . . . . . . . . . . . . . . . . 150

6.3

A plate with PCLD treatment under CFCF boundary conditions 151

6.4

Schematic of sensor array for plate testing . . . . . . . . . . . . . 151

6.5

Contour plot of experimental bending mode shape functions for

. . . . . . . . . . . . . . . . . . 150

an aluminum plate with CFCF boundary conditions . . . . . . . . 169 6.6

Contour plot of analytical bending mode shape functions for an aluminum plate with CFCF boundary conditions . . . . . . . . . 170

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6.7

Frequency response functions of an aluminum plate with CFCF boundary conditions, at location 15, as shown as Table 6.1; in which only 7 plate modes were included and 25 beam bending modes were used . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.8

Contour plot of experimental bending mode shape functions for a plate with PCLD treatment under CFCF boundary conditions . . 172

6.9

Contour plot of analytical bending mode shape functions for a plate with PCLD treatment under CFCF boundary conditions . . 173

6.10 Frequency response functions of a plate with PCLD, at location 11; in analysis I, 25 modes for each displacement were assumed and it leads to 500 degrees of freedom; in analysis II, 16 modes for each displacement were used for a total of 320 degrees of freedom. 174

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Chapter 1

Introduction

1.1

Motivation and Objectives

Damping augmentation of structures is of key interest to aerospace, mechanical, and civil engineers. Noise and vibration reduction is a major challenge pertaining to these fields; especially in aerospace applications, such a reduction must be achieved with a minimal increase in weight. Viscoelastic damping layers integrated within the structures of vibrating members have been used towards this end. The current research work is motivated by the helicopter industry’s goal of achieving a “jet smooth quiet ride,” in which smart materials and structures technologies may be exploited and applied. Our research task is the hybrid/active trim panel damping control in reduction of the interior cabin noise of the helicopter. Two primary contributions to the interior noise are airborne noise and structure-borne noise. Airborne noise, which occurs mainly in a low frequency

1

range below 500 Hz, is due to main and tail rotors. Structure-borne noise is primarily responsible for frequency components in the range from 500 Hz to 6000 Hz. The transmission adds these higher frequency disturbances, and these disturbances are borne through the air and via structural vibration into the cabin. Three different approaches to control noise have been taken in the past: 1) passive scheme, 2) active scheme, and 3) hybrid scheme. Passive schemes include increasing damping material and stiffening of the structures [47]. In this approach, damping and stiffness characteristics of a structure are enhanced. Aircraft skin with damping tape is effective for reducing the response of the skin, and consequently, noise transmission in the high frequency range. The damping tape is not effective at a low frequency range (below 500 Hz). Therefore, passive control approaches are ineffective, particularly for low frequency. Also an undesirable consequence of passive damping method in aerospace applications is an increase of weight, which penalizes performance. The second approach involves actively controlling interior noise using secondary sources. The active schemes use secondary acoustic sources as in Nelson and Elliott [48] or secondary force actuators as in Fuller and Jones [22] and Balachandran et al [2]. The stateof-the-art for the active control scheme has been recently reviewed by Hansen [26]. Hansen described the currently available control system hardware, software, and control sources. Active control schemes overcome the weight penalties, but these method are effectively limited to low frequency bands, less than 500 Hz.

2

The third scheme for noise control is a hybrid scheme. This scheme utilizes features of both passive and active control schemes, in which active control can handle low frequency and passive control can handle high frequency disturbances. One such hybrid approaches is the concept of active constrained layer damping (ACLD) scheme by Baz [4] and its application to noise control by Poh et al [51]. In our previous research, we proposed a hybrid scheme [59, 60] with the use of viscoelastic material for passive damping augmentation, and a surface piezo patch actuators for active control. The passive dissipative layers can damp out higher frequency components of the disturbances, while control of the actuators through control algorithms can suppress noise in the lower frequencies. A preliminary study of this hybrid scheme was investigated by Veeramani [60]. The structures considered were a three-layered sandwich plate, which were used to emulate a trim panel. A viscoelastic core was sandwiched between the two face layers of the trim panel and the viscoelastic core has a frequency dependent complex shear modulus. The assumed mode method was used to solve the system using beam and rod mode shape functions. Experiments were conducted for the case where all edges of the sandwich plate were clamped. Natural frequencies and loss factors predictions were validated experimentally. We continue this task to further study the sandwich beams and plates in order to obtain more accurate solutions. Therefore, the objectives of this research are: 1. To develop a spectral finite element method using higher order interpolation

3

functions to reduce computational cost for sandwich beam analysis; 2. To implicitly account for frequency dependent stiffness and damping of viscoelastic materials by using the spectral finite element method in sandwich beam analysis. A corollary of this is that we do not want to add internal dissipation coordinates which increase degrees of freedom, and hence the computational cost; 3. To directly solve for the bending and in-plane mode shape functions of isotropic rectangular plate vibration based on the Kantorovich method and utilize these mode shape functions in the assumed modes method for the sandwich plate analysis to reduce the computational cost compared to the previous work which used 1D beam and rod modes; 4. To validate natural frequencies, damping, mode shapes and response experimentally in all of the above cases. The following section describes the state-of-the-art in the models of viscoelastic material, sandwich beam and plate and solution types.

1.2 1.2.1

State-of-the-art Viscoelastic Materials

Fundamental damping concepts and methods to characterize damping are presented in the book by Nashif et al [47]. Nashif et al represented viscoelastic

4

materials using a complex modulus in the frequency domain. When excited by a harmonic force of constant amplitude, the steady state response of a simple single-degree-of-freedom system can be used to determine the damping through the response amplitude at resonance, Nyquist plots, hysteresis loops and dynamic stiffness. Thus a frequency dependent complex modulus can be determined, that experimentally captures the steady state behavior of viscoelastic materials to sinusoidal excitation. The temperature nomogram was developed by Jones [29] to represent such data in a master curve that is convenient for practical applications. Several of these data sheets are shown in appendices of Nashif et al [47], as well as in manufacturer data sheets, for example 3M [56]. All the damping models for the viscoelastic materials must capture the frequency dependent complex modulus in the frequency domain and demonstrate the creep and relaxation properties in the time domain as well. Traditional damping models were reviewed by Sun and Yu [58]: the Maxwell, the Kelvin, and the Zener model [71]. However, these models have drawbacks and cannot capture the real behaviors of the viscoelastic materials [47, 58]. Creep functions predicted by the Maxwell model and relaxation functions predicted by the Kelvin model are unrealistic for the viscoelastic materials. The Zener model can predict both creep and relaxation functions well but it failed to capture the curve of frequency dependent complex modulus in the frequency domain realization. Christensen [12] discussed the viscoelasticity theory, in which a time domain model using a relaxation function was developed. This time domain model can

5

be transformed into the frequency domain, thereby gaining a complex modulus. The properties of the relaxation function were discussed based on the physical principles. But it is very difficult to find such relaxation functions to capture a complex modulus in the frequency domain through transformations. He did not present a relaxation function as an example either. Bagley and Torvik [3] tried to improve the traditional damping models as discussed above, but it turned out to be a very complicated frequency model because there are five parameters have to be determined by curve fitting the experimental data. It is also difficult to transform the model into the time domain because it involves complicated assembles of the system matrices. Recent damping models were developed in order to curve fit this master curve of the viscoelastic material, that is, the complex modulus as a function of frequency. Instead of deriving the damping force, these models introduce additional internal dissipation coordinates to curve fit the complex modulus in the frequency domain and transform back to time domain. The GHM method [23, 42, 43] was developed using mini-oscillators. This model can be easily incorporated into conventional finite element or other analyses to account for the frequency dependent complex shear modulus. There are other models such as the ATF and ADF method [37, 38, 39] and Yiu’s model [69, 70]. These modern damping models can be used to account for the frequency dependent complex shear modulus of viscoelastic materials. In our research, the GHM method was adopted and incorporated in the conventional finite element method or the

6

assumed modes method for the sandwich beam and plate analyses. These additional dissipation coordinates increase the size of problem and lead to a large degree of freedom model. In order to mitigate the computational cost, we need to develop a method in the frequency domain to implicitly account for the frequency dependent complex shear modulus of viscoelastic materials for the analyses of sandwich structures.

1.2.2

Sandwich Beams

For the surface damping treatments, Sun and Yu [58] summarized prior research. There are two types of surface damping treatment: unconstrained, and constrained, layer treatments. For the unconstrained layer treatment, a layer of viscoelastic tape is applied to the surface of a host structure. The energy is dissipated by the cyclic tensile and compression strain when the host structure is in bending motion. For the constrained layer treatment, a stiff layer is added to the top surface of the viscoelastic layer. When the sandwich structure undergoes bending motion, this constraining layer causes a significant shear deformation in the constrained, or sandwiched viscoelastic layer, so that the energy can be dissipated. The constrained layer damping treatment is more effective because the viscoelastic materials dissipate energy mainly by shear deformation and the constraining layer enhance the magnitude of shear deformation. Sandwich beam, plate, and shell structures, as reviewed in [58], have been developed in practical

7

applications for damping augmentation. Kerwin [31] presented the first analysis of the simply supported sandwich beam using a complex modulus to represent the viscoelastic core. His model predicted attenuation of a traveling wave on either a simply supported or infinitely long beam. DiTaranto [16] extended Kerwin’s work by developing a sixth order differential equation of motion in terms of the longitudinal displacement. Mead and Marcus [44] derived the same order differential equation of motion in terms of transverse motion of sandwich beam and presented wave propagation solutions. Both works used the Kerwin’s basic assumptions, in which the viscoelastic core has a complex modulus and the energy is dissipated by the shear deformations in the viscoelastic core, and both extended Kerwin’s work by allowing for more general boundary conditions. Since the 1950s, there have been many papers published on the theory and application of constrained layer damping. Many researchers used Kerwin’s assumptions, and investigated the validity of assumptions, and damping or loss factor predictions. Closed-form solution methods were typically used because finite element techniques were not readily available for this class of problems. Nakra [46] and Mead [45] reviewed all this area and they discussed the differences and similarities between the theories. The above theories laid the foundation for the analysis of sandwich beams with constrained layer damping treatments. Douglas and Yang [18, 19] studied the partial and fully passive constrained layer damping (PCLD) treatment for beam structures. Experiments were con-

8

ducted to obtain the responses, which were compared to predictions based on a progressive wave solution method. They considered two kinds of damping mechanism in a sandwich beam structure. One was the shear damping in the viscoelastic core which was due to the shear deformation as discussed before. The other was the compression damping in the viscoelastic core. When there was relative transverse motion in the constraining layer and base beam structure, the viscoelastic core undergoes compression to dissipate energy. They concluded that shear damping is a broad band mechanism for most engineering purposes. The compression damping in the viscoelastic core must be considered only within a certain spectral band. The damping of transverse dynamics of the thin sandwich beam is outside the spectral influences of compression damping. Therefore, the shear damping mechanism was mainly considered in thin sandwich structures. In 1982, Johnson et al [28] published a work on using modal strain energy (MSE) methods for damping design by finite element methods. The available FEM packages enable us to obtain the numerical solutions of sandwich structures much easier. But for the sandwich beam, two-dimensional finite elements were used to model essential one-dimensional beam like structures. Baz [4] first replaced the constraining layer by piezo-ceramic material to develop the active constrained layer damping (ACLD) treatment. The ACLD treatment were rapidly adapted in structural vibration control [5, 34, 39, 40, 57]. There are two classes of methods to evaluate the beam with PCLD or ACLD treatment, that is the assumed modes method (AM) in [34, 40] or conventional fi-

9

nite element method (CFEM) [39, 49], Additional damping models using internal dissipation coordinates are incorporated to account for the frequency dependent complex modulus. However, introduction of internal dissipation coordinates will greatly increase the size of the numerical problem. For example, if the total degrees of freedom of a sandwich beam were N and three mini-oscillators were used in the GHM method, this will lead to a system with total degrees of freedom 4N. An effective and accurate method is needed to analyze sandwich beams with the PCLD or ACLD treatments, that implicitly account for the frequency dependent complex modulus of the constrained viscoelastic layer, without the addition of internal dissipation degrees of freedom.

1.2.3

Sandwich Plates

Ross et. al. [54] studied simply-supported plates, and assumed a perfect interface and compatibility of transverse displacement in each layer. Rao and Nakra [52] [53] developed the basic equations of vibratory bending of asymmetric sandwich plates with isotropic face-plates and viscoelastic core. Lu et. al. [41] developed a finite element model and presented experimental data for sandwich plates under free boundary conditions. Cupial and Niziol [15] used the variational method to model sandwich plates with anisotropic face-plates, who presented simplified forms of the equations for a symmetric plate or for specially orthotropic face

10

layers. The modal frequencies and modal loss factors predicted by the analysis were compared well with the results in Johnson and Keinholz [28]. However, they did not present experimental validation for the modal frequencies and loss factors. Baz and Ro [6] studied plates with active constrained layers for vibration control and a two-dimensional finite element model was developed to model the sandwich plate structures. Experiments were conducted to show the response of the sandwich plate with or without controller. Veeramani [60] followed Cupial and Niziol’s work and developed the models for the sandwich plates with surface bonded piezo-ceramic actuators. The face layers in the sandwich plate are assumed to be anisotropic material and the viscoelastic core is assumed to have frequency dependent complex shear modulus. Experiments were conducted to test three sandwich plates with the isotropic face plates. The assumed modes method was used to analyze the sandwich plate system using beam and rod modes. More modes, especially in-plane modes, must be included in order to achieve good frequency predictions compared to experimental data because these modes need to capture the shear deformation in the viscoelastic core. Wang et al [63] [64] improved the analyses to include the GHM method to account for the frequency dependent complex shear modulus of the viscoelastic core. The number of in-plane mode is still large to obtain the comparable frequency solutions. Instead of resorting to FEM package, an improved assumed modes method is needed to better predict the natural frequency, loss factors, mode shapes, and responses of sandwich plates. Experimental data for these are needed as well.

11

1.3

Scope of the Present Research

Based on the literature reviews, sandwich beams and plates have been well studied. However, improving the effectiveness and accuracy of solutions for those structures is still an important goal. We seek to develop higher order analyses of the sandwich beams and plates using wave solutions and plate mode shapes. For sandwich beam structures, we try to implicitly account for the frequency dependent complex shear modulus of the viscoelastic core and develop a spectral finite element method based on a progressive wave solutions in the frequency domain. For sandwich plate structures, we try to update the beam and rod modes used in the assumed mode method using plate mode shape functions which were solved from bending and in-plane vibration of an isotropic rectangular plate based on the Kantorovich method. Experiments will be conducted to validate our analytical results for both sandwich beams and plates.

1.3.1

Sandwich Beam

To have sufficiently accurate higher mode number natural frequency estimates for sandwich beam structures incorporating viscoelastic damping layers, a very large number of degrees of freedom are required. This large number of degrees of freedom is represented by the sum of the number of elements or assumed modes needed for accuracy at higher frequency, plus the additional internal dissipation coordinates that must be added to each element or assumed mode to account for

12

the frequency dependent properties of the viscoelastic core. We are especially interested in developing a method that alleviates the large number of degrees of freedom needed to analyze sandwich beams using CFEM or AM coupled with internal dissipation coordinate methods such as GHM [23, 42] or ADF [38]. One possible approach was developed by Douglas [18] based on the governing equation given by Mead and Marcus [44]. The progressive wave solution was used to calculate frequency response functions using an impedance matrix. Douglas was the first to explore wave solutions in order to implicitly account for the frequency dependent complex modulus of viscoelastic core in the solution method. However, the disadvantages of the progressive wave method described by Douglas [18] are that if the boundary conditions or structural junctions (joints, change in cross-section, etc.) change for a structure, the impedance matrix must be rederived. The primary advantages of the progressive wave solution are: (a) the method implicitly accounts for frequency dependent complex modulus of the viscoelastic core without adding internal dissipation coordinates, (b) the method solves for the frequency response directly from the governing equation without resorting to the modal expansion or displacement interpolation functions of AM and CFEM methods, respectively. The primary goal is to develop a finite element method in the frequency domain, based on wave propagation solutions, that mitigates the disadvantages of Douglas’ progressive wave solution. A spectral finite element (SFEM) methodology was formulated for isotropic structures by Doyle [17]. This methodology can be extended to analyse sandwich

13

beams with isotropic face layers and a viscoelastic core. Baz [7] used a spectral finite element model to describe the longitudinal waves in rods treated with active constrained layer damping. We [62] developed a spectral finite element model for a beam with PCLD treatment. Kim and Lee [32] also applied the spectral finite element for the beam with ACLD treatment. Both spectral finite element methods for sandwich beam analyses were developed simultaneously and presented at same session of AIAA Structures, Structural Dynamics, and Materials Conference. A small number of elements, as compared to CFEM methods, was needed to calculate the frequency response function. To obtain accurate results using the SFEM method, only as many elements are needed as there are junctions between substructures of different impedance. Thus, the SFEM method requires that a much smaller number of degrees of freedom be incorporated in the solution. In contrast to the assumed modes and conventional finite element methods, SFEM can directly handle the frequency dependent complex modulus of the viscoelastic core, without adding any internal dissipation coordinates. Because the SFEM calculates an impedance matrix at each frequency of interest, the complex shear modulus can be adjusted at each frequency of interest as well. The shape functions used in SFEM are based on the exact displacement of wave solutions, as opposed to the polynomial interpolation functions typically used by conventional FEM. We will present details of a SFEM analysis of the flexural vibration of sandwich beams with a viscoelastic core. The frequency response

14

functions (FRFs) are calculated using SFEM and compared to those computed using CFEM analyses and validated by the experimental results.

1.3.2

Sandwich Plate

In our previous work [64], the assumed modes method was successfully applied to the analysis of sandwich plates with isotropic face plates and a viscoelastic core using beam and rod mode shapes. The modal frequencies were calculated and validated by experiment by considering a plate with all four sides having clamped boundary conditions. The numerical predictions agreed well with experimental solutions. In our previous analysis, rod and beam mode shapes were only approximations of in-plane and bending mode shapes in x and y direction, respectively. These mode shapes are only admissible functions and do not satisfy the plate vibration governing equations. As discussed in our work, we need to include more mode shapes, especially in-plane plate mode shapes to achieve comparable accuracy compared to experimental data. The first 25 in-plane and bending plate mode shapes, which were approximated by rod modes and beam bending modes, were included. The large number of mode shapes plus the internal coordinates in GHM method increase the size of the problem and add the computational cost. Our objective is to alleviate the computational cost by using fewer assumed mode shapes. We will update the in-plane and bending mode shapes by directly

15

solving for the mode shapes of isotropic plate in-plane and bending vibrations using the Kantorovich variational method [30]. The Kantorovich variational method shows the equivalence of solving boundary values problem of partial differential equations (PDEs) and finding functions to minimize the integral of associated total system energy. This equivalence enable us to obtain the analytical solutions of mode shapes for in-plane and out-of-plane plate vibrations. An iteration scheme was developed to calculate the natural frequency and corresponding mode shapes. In the Kantorovich method, we will solve a series of coupled ordinary differential equations by assuming separable expression of solutions. Bhat et al [8] solved plate bending mode shapes for boundary conditions of combinations of simply-supported and clamped cases. We follow the same method to solve out-of-plane (bending) plate mode shapes with free or clamped boundary conditions. The analytical expressions for bending mode shapes are described. On the other hand, few papers discuss mode shapes of in-plane plate vibration. Recent, Farag and Pan [20, 21] presented frequency results of in-plane plate vibration. They solved the governing equations by using rod mode shapes and an iteration scheme is used to obtain the modal frequencies. In our analysis, the Kantorovich method is applied to in-plane plate vibration problems [65]. Our results are validated by Farag and Pan [21]. We will outline this method and show how to obtain the mode shapes for plate in-plane and bending vibrations. These mode shapes are used to analysis sandwich plate by the assumed modes

16

method. The new two-dimensional plate mode shapes will show improved computational efficiency because fewer number of mode shapes is needed to achieve same accuracy as compared to those results using one-dimensional beam and rod mode shapes. This leads us a higher order method for sandwich plate analysis.

1.4

Organization

This dissertation is organized as follows. Chapter 2 reviews the existing damping models for the viscoelastic materials and demonstrates the advantages and drawbacks for each model. Chapter 3 discusses the analyses of sandwich beam. The spectral finite element method is presented in details. The assumed modes method and conventional finite element method are outlined as well. All the analytical results are validated by experimental data. Chapter 4 sets up the assumed mode analysis for the sandwich plate. The results of natural frequency and loss factors are presented to compare with the previous experimental data. A parametric study of temperature effects on the complex shear modulus is presented. We demonstrate that the natural frequency and loss factors vary with the change of the temperature. The temperature issue is a big concern in the design of the sandwich structures. Chapter 5 presents the Kantorovich method for the plate bending and in-plane vibrations. The mode shape functions are calculated and given in the closed-form solutions. Chapter 6 presents the updated assumed modes method using plate modes for the analyses of sandwich plate.

17

Experiments are conducted to validate the results of bending frequency, mode shape function and response for an aluminum plate, and to validate the results of natural frequency, loss factor, mode shape function and response for a sandwich plate with partial PCLD treatment. Finally, the conclusions are presented in the Chapter 7. The mass and stiffness matrices in the assumed modes method for the sandwich plate analyses are present in the Appendix.

18

Chapter 2

Viscoelastic Materials

This chapter presents dynamic characteristics of viscoelastic damping materials and outlines the different existing mathematical models. Classical representations of viscoelastic materials include the Maxwell model, the Kelvin-Voigt model, and the Zener model (Standard Solid Model). We will show that these models cannot capture the behavior of viscoelastic damping materials. Our efforts then focus on some of the modern models including: the Fractional Derivatives (FD) method [3], the complex modulus [47], the Golla-Hughes-McTavish (GHM) method [23], the Augmenting Thermodynamic Fields (ATF) method [37], and the Anelastic Displacement Field (ADF) method [38]. The merits and limitations of the newer methods are also presented. The GHM method was used primarily in this research because it can be easily adapted to the traditional techniques of structural analyses.

19

2.1

Characteristics of Viscoelastic Materials

Viscoelastic damping is exhibited in many polymeric and glassy materials and this internal damping mechanism is very important for damping augmentation to reduce vibration and noise in structures. The damping arises from relaxation and recovery of the polymer network after it has been deformed. Because viscoelastic materials exhibit both viscous and elastic characteristics, they hold unique properties. For example, in addition to undergoing an instantaneous displacement, when subjected to a constant force, they also undergo creep over a period of time. Alternatively, the force required to maintain a given deformation decreases over a period of time. This phenomenon is called relaxation. The stress and stain constitutive relationship for a linear viscoelastic material is: σ(t) = Eǫ(t) +


t

g(t − s)

dǫ(s) ds ds

(2.1)

0

where σ(t) is the stress and, ǫ(t) is the strain and the kernel function g(t − s) is known as a relaxation function. The relaxation function is the stress response to a unit-step strain input. Christensen [12] discussed the expected characteristics of the relaxation function based on both reasonable hypotheses and thermodynamic considerations. This linear hereditary stress and strain law can be expressed in the Laplace domain. The Laplace transformation of the above equation yields ˜ ǫ(s) σ˜ (s) = sE(s)˜

20

(2.2)

Substituting s = jω into above equation yields a complex modulus ˜ E ⋆ = E ′ (ω) + jE ′′ (ω) = jω E(jω)

(2.3)

where E ′ (ω) is the storage modulus and E ′′ (ω) is the loss modulus. The loss factor is defined as the non-dimensional quantity obtained by dividing the imaginary part of the complex modulus by the real part: η(ω) =

E ′′ (ω) E ′ (ω)

(2.4)

The loss factor measures the average ratio of energy dissipated from the viscoelastic material per radian to the maximum stored energy under a sinusoidal force. A complex modulus that describes the steady state response of the viscoelastic material to a sinusoidal load is used to capture the characteristics of viscoelastic materials. The complex modulus is dependent on the steady state frequency of harmonic excitation as well as temperature. There are various techniques for determining the complex modulus of viscoelastic materials experimentally. The details of test setup, specimen selection criteria, test procedures and other relevant issues are discussed by Nashif el at [47]. The important characteristics of damping materials are presented in a temperature nomogram. The temperature nomogram was developed by Jones [29] and is considered a standard graphical presentation of complex modulus data. The data for the nomogram of viscoelastic materials are usually provided by manufacturers. The storage shear modulus and loss factors can be read from the nomogram. In the nomogram, a vertical

21

scale for frequency (Hz) is on the right, the shear modulus (GPa) as well as loss factor are on the left, and the diagonal lines represent the temperature. Using the nomogram, a complex shear modulus can be determined for a certain frequency as well as temperature. In Figure 2.1 is presented the nomogram for 3M Scotchdamp ISD112, a typical viscoelastic material. The complex shear modulus data can be read from the nomogram, shown in Figure 2.2, in which the storage modulus, G′ , and loss factors, η, are plotted versus the frequency at temperature of 20o C.

2.2

Classical Damping Models

Usually in engineering applications, a phenomenological approach is taken to model viscoelastic behavior. There are three classical mathematical models for viscoelastic damping. As mentioned in Section 2.1, elastic-type behavior of viscoelastic damping materials refers to an instantaneous incremental proportionality of stress and strain. Visco-type behavior of viscoelastic materials infers “dashpot motion”, where a component of stress is proportional to the rate of change of strain. Therefore, a number of simple one dimensional models of viscoelastic materials are based on combinations of spring and dashpot elements to represent the elastic and visco-type of motions respectively. These models include the Maxwell, Kelvin-Voigt, and Zener (Standard Solid) models. We first present the mathematical representations of all three models and then demon-

22

strate drawbacks of each model. • The Maxwell model is presented in terms of serial combination of a viscous damper and an elastic spring as shown in Figure 2.3(a). The relationship of the stress and strain is σ(t) +

dǫ(t) cd dσ(t) = cd Es dt dt

(2.5)

• The Kelvin-Voigt model is presented in terms of a parallel combination of a viscous damper and an elastic spring as shown in Figure 2.3(b). The relationship of the stress and strain is σ(t) = Es ǫ + cd

dǫ(t) dt

(2.6)

• The Zener model is presented in terms of a serial and parallel combination of a viscous damper and two elastic springs, as shown in Figure 2.3(c). The relationship of the stress and strain is σ(t) + α

dσ(t) dǫ(t) = Eǫ + Eβ dt dt

(2.7)

The spring constants Es and Ep , act as Young’s modulus, and cd is a damping coefficient. The parameters α, β, and E are defined as: cd Es + Ep cd β = Ep Es Ep E = Es + Ep α =

23

(2.8) (2.9) (2.10)

As discussed in Section 2.1, viscoelastic materials have two unique properties in the time domain: creep and relaxation. Figure 2.4 shows the time history of the creep functions of the three models and Figure 2.5 shows the time history of the relaxation functions of the three models. As we know, the characteristic of the creep function should increase with time and converge to a final value in steady state and the characteristics of the relaxation functions should decrease with time and reach a final value in steady state. We note that the creep function predicted by the Maxwell model and relaxation function predicted by the KelvinVoigt model are unrealistic. The creep function predicted by the Maxwell model keeps increasing with time and the relaxation function predicted by the KelvinVogit model keeps constant with time. Therefore, both the Maxwell and KelvinVoigt models fail to capture time domain characteristics of viscoelastic materials. On the other hand, the Zener model can predict both creep and relaxation functions well in the time domain. We need to access the realization of the Zener model in the frequency domain as well. We assume harmonic responses of stress and strain as follows: σ = σ0 ejωt

(2.11)

ǫ = ǫ0 ejωt

(2.12)

Then substituting Eqs. 2.11 and 2.12 in to Eq. 2.7 yields σ0 = E ′ (1 + jη)ǫ0

24

(2.13)

where E(1 + αβω 2) 1 + α2 ω 2 (β − α)ω η = 1 + αβω 2

E′ =

(2.14) (2.15)

The above equations show that the complex modulus is a function of frequency which reflects some aspects of real viscoelastic behavior. However, when we compare the Bode plot of E ′ and loss factor η to experimental data, the variation of E ′ and η with frequency is much more rapid than that observed in experimental data. Therefore, the Zener model is only an approximation and needs to be improved to match the experimental data. The limitation of the Zener model can be reduced by including additional derivatives of σ and ǫ, as follows: σ(t) +

 k

αk

 dk ǫ(t) dk σ(t) = Eǫ + E βk dtk dtk

(2.16)

k

where k is an integer. This generalized standard solid model improves the Zener model but a drawback of this improved model is that a substantial number of terms are needed to capture the viscoelastic properties over a wide frequency range.

2.3

Modern Damping Models

The complex modulus of viscoelastic materials was developed in the frequency domain and can be experimentally measured in a straightforward fashion. If we

25

can develop our structural analysis techniques in the frequency domain, we can take advantage of experimental data directly and implicitly, there is no need to develop explicit damping models. Otherwise, we must incorporate a time domain damping model that can account for the frequency dependent complex modulus. Structural analysis techniques are mainly developed in the time domain, so a time domain representation of complex modulus is a must. Since the 1980s, damping models have been developed that are based on the phenomenological method. We outline some of these modern damping models for viscoelastic materials and address the merits and limitations of each model separately.

2.3.1

Fractional Derivatives Model

Bagley and Tovik [3] developed a fractional derivative model for describing viscoelastic behavior. This model was motivated by reducing the number of terms in the generalized standard solid model as discussed in Section 2.2. The representation of the complex modulus of viscoelastic materials in Laplace domain is: E0 + E1 sα E = 1 + bsβ ⋆

(2.17)

There are five parameters, E0 , E1 , b, α and β, which are used to in order to curve fit the experimental data. The advantage of this model is that it closely fits the experimental data over a significant range of frequency. The major drawbacks include the awkward assembly of the global equations of motion and

26

the large cumbersome system matrices produced. This model is good only in the frequency domain, because taking the inverse Laplace transform of a frequency domain representation of complex modulus based on a fractional derivative is difficult, and can be done in an approximate way.

2.3.2

AFT and ADF Models

Some of the shortcomings of frequency models can be overcome by using time domain representation of viscoelastic materials. A number of approaches have been developed to account for the frequency dependent properties of viscoelastic materials while also providing time domain analysis. These include the GollaHughes-McTavish (GHM) method [23], the Augmenting Thermodynamic Fields (ATF) method [37], and the Anelastic Displacement Field (ADF) method [38]. Instead of deriving damping force, these methods use additional internal dissipation coordinates to account for the frequency dependent complex modulus. These methods can be easily incorporated into finite element models that are capable of predicting dynamic response of a structure with viscoelastic material. The Augmenting Thermodynamic Fields (ATF) method was developed by Lesieutre and Mingori [37]. It is a time domain continuum model of material damping that preserves the characteristic frequency dependent damping and modulus of real materials. Irreversible thermodynamics were used to develop coupled material constitutive relations and partial differential equations of evo-

27

lution. These equations are implemented in a numerical solution of the finite element method. The details of the model were presented by Lesieutre and Mingori [37]. The ATF method describes the interaction of the displacement field with irreversible processes occurring at the materials level, while the Anelastic Displacement Field (ADF) method by Lesieutre et al [38, 39] focuses on the effects of such processes on the displacement rather than the process themselves. The total displacement field is considered to consist of two parts: an elastic part and an anelastic part. The ATF method provides a good physical explanation of viscoelastic damping mechanism in the structure, while the ADF method leads to straightforward finite element solutions because the anelastic displacement fields are similar to the elastic displacement fields. Because both ATF and ADF methods lead to a first order damping model, only state space forms can be used when combined with structural analytical models. An unfortunate consequence of using this method is that the global mass matrix is singular.

2.3.3

Golla-Hughes-McTavish Model

The Golla-Hughes-McTavish (GHM) method is a technique for deriving viscoelstic finite elements from commonly used elastic finite elements and measurements of frequency dependent complex modulus. Auxiliary dissipation coordinates,

28

which are internal to each viscoelastic element, permit general description of frequency dependent viscoelastic material properties via the mini-oscillator as shown in Figure 2.6. The effect of the mini-oscillator includes a second order rational function involving three parameters. These mini-oscillators are used to curve fit the experimental data in the frequency domain, and a time domain representation of viscoelastic model can be achieved through Laplace transformation. The mini-oscillator mechanical analogy, as shown in Figure 2.6, is an equivalent system with respect to the dynamic stress and strain behavior of viscoelastic material associated with the displacement q. The dissipation coordinate z appears as an augmenting state variable which has no direct physical significance. The mass term of the mini-oscillator does not represent a real mass in the structural system and does not contribute to the kinetic energy. The mass, spring and damper system is used to represent the behavior of the viscoelastic material compared to spring and damper combinations in the Zener model as discussed in Section 2.2. The inertial effects due to the introduction of mass in GHM method help us capture the slower change of complex modulus with frequency. This is why the GHM method is very accurate compared to general standard solid Zener model. In the GHM method, the complex shear modulus is written in the Laplace

29

domain as: 

G⋆ = G0 1 +

N  k

s + 2ζˆk ω ˆk s α ˆk 2 ˆ s + 2ζ k ω ˆks + ω ˆ2 2

k



(2.18)

where G⋆ is the complex shear modulus of the viscoelastic material, and the factor G0 is the equilibrium value of the modulus, i.e. the final value of the relaxation function, and s is the Laplace domain operator. The parameters are obtained from the curve fitting to the complex modulus data for a particular viscoelastic material (Lam, Inman and Saunders, 1997). The number of terms, N, retained in the expression is determined from the high or low frequency dependence of the complex modulus. For example, in order to capture the complex shear modulus in the frequency range from 1 Hz to 500 Hz of the viscoelastic materials shown in Figure 2.2, three mini-oscillators were used in the curve fit. The residual of the optimization is of the order 10−2 over the desired frequency range. The results of storage modulus and loss factors which were predicted by the GHM method were plotted versus frequency and compared to experimental data in Figure 2.7. The parameters used in three mini-oscillators are: G0 = 1.0e5     = ˆ2 α ˆ3 1.59 6.6 32.0 α ˆ1 α     = ˆ2 ω ˆ3 1.0e4 2.0e4 0.5e4 ω ˆ1 ω     = 348.8 56.4 1.0 ζˆ1 ζˆ2 ζˆ3

(2.19) (2.20) (2.21) (2.22)

The time domain relaxation function found using the GHM method can be

30

expressed as: 

G(t) = G0 1 +

N 

−b1k t

αk

b2k e

k

−b2k t

− b1k e b2k − b1k



(2.23)

where b1k , b2k = ω ˆk



ζˆk ∓

 2 ˆ ζk − 1

(2.24)

The time domain function of G(t) exponentially decays with time t if b1k and b2k are distinct real constants and captures the relaxation properties of the viscoelastic material. We have shown that the mini-oscillator is used in the GHM method to equivalently represent the behavior of viscoelastic material. The parameters in the mini-oscillator term are used to curve fit the experimental data of the storage modulus and loss factor versus frequency and temperature. After the parameters are determined, the GHM method is incorporated into conventional dynamic structural analytical techniques for structures with viscoelastic components. In general, the structural dynamics equation can be expressed in the Laplace domain as: ¯ Ms2 q(s) + Ke q(s) + G⋆ Kq(s) = f (s)

(2.25)

where, M is a mass matrix, Ke is a stiffness matrix contributed from elastic components in the structures, and G⋆ is a complex shear modulus of a viscoelastic material. We assume there is only one viscoelastic material on the structure. An auxiliary coordinate z is introduced zk (s) =

ω ˆ k2 x(s) s2 + 2ζˆk ω ˆks + ω ˆ2 k

31

(2.26)

Using this new dissipation coordinate, Equation 2.25 can be rewritten as:     

0 0  2  0   M s +   s+      ˆ 0 α ˆ 2ωˆζ K0 0 α ˆ ωˆ12 K0     

ˆ 0 −ˆ αK0   q(s)   f   Ke + K0 + αK =           0 z(s) −ˆ αK0 αK ˆ 0

(2.27)

(2.28)

The Laplace domain expression of the governing equation has a second order time domain realization: 











0 0  M   q¨   0   q˙     +          ˆ 0 α ˆ ωˆ12 K0 0 α ˆ 2ωˆζ K0 z¨ z˙     

where

(2.29)

ˆ 0 −ˆ αK0   q   F   Ke + K0 + αK    =  +      0 z −ˆ αK0 αK ˆ 0 ¯ K0 = G 0 K

(2.30)

We demonstrate the case for only one mini-oscillator, N = 1. Because K0 is usually positive semi-definite, the above mass matrix may not be positive definite. To remedy this situation, spectral decomposition of K0 is used, as suggested by McTavish and Hughes [42] ¯ = G0 R ¯Λ ¯R ¯T K0 = G 0 K

(2.31)

¯ is a diagonal matrix of the nonzero eigenvalues of K ¯ and the columns where Λ ¯ correspond to orthonormalized eigenvectors. The above case of a single of R

32

mini-oscillator term can be easily extended to a multi-oscillator model. The general form of the GHM method [42] is formed using the stiffness matrix, K; ¯ are and damping matrix, D; and mass matrix, M   ˆ −ˆ α1 R  Ke + K0 (1 + α)    −ˆ α1 RT α ˆ1 Λ   K= ..  . 0    −ˆ αn RT 0 

     ¯ M =      

where

     D=     

given by: 

· · · −ˆ αn R    0 0     .. . 0     0 α ˆn Λ

M

0

···

0

0

α ˆ 1 ωˆ12 Λ

0

.. .

.. .

0

..

0

···

0

0

1

0

.

0 α ˆ n ωˆ12 Λ n

···

0

0 α ˆ 1 2ωˆζ11 Λ

0

.. .

.. .

0

..

0

···

0

ˆ

.

0 ˆ

α ˆ n 2ωˆζnn Λ

(2.32)

           

(2.33)

           

(2.34)

¯ Λ = G0 Λ ¯ R = RΛ Finally, we obtain the constant mass, damping, and stiffness matrix for the structure with viscoelastic materials. The size of our original problem increases

33

because of the introduction of dissipation coordinates that are internal degrees of freedom. The GHM method has been successfully applied to the conventional dynamic analyses for structures with viscoelastic materials.

2.4

Summary

In this chapter, we showed that the frequency dependent complex modulus method can be used to represent viscoelastic materials. Experimental data are presented in the nomogram, which is a master curve and the complex shear modulus of viscoelastic materials are functions of frequency and temperature. We reviewed three classical damping models, that are the Maxwell model, the Kelvin model, and the Zener model. They cannot be applied to viscoelastic damping materials, because they fail to capture the behavior of the viscoelastic materials. Some modern damping models were discussed as well. They are the fractional derivative (FD) method, the Augmenting Thermodynamic Fields (ATF) method, the Anelastic Displacement Field (ADF), and the Golla-HughesMcTavish (GHM) method. The FD model is a good model only in the frequency domain. In the AFT, the ADF, and the GHM method, the additional internal dissipation coordinates were used to account for the frequency dependent complex modulus of viscoelastic materials. We adopted the GHM method in this research, which is most applicable to structural analysis. But we need to develop methods that implicitly account for the frequency dependent complex modulus

34

without adding internal dissipation coordinates which increase degrees of freedom of the system.

35

Figure 2.1: Nomogram of the viscoelastic material, 3M ISD 112.

7

10

6

10

5

10

4

10

Storage Modulus 3

10

2

Loss Factors

10

1

10

0

10

−1

10

0

10

1

2

10

10

3

10

Frequency: Hz

Figure 2.2: Storage Modulus and Loss Factor Vs. frequency at temperature 20o C for the viscoelastic material 3M ISD 112.

36

Es

Cd

(a) Maxwell Model

Es

Cd

(b) Kelvin-Voigt Model Es Ep Cd

(c) Zener Model

Figure 2.3: Classical models of viscoelastic materials

ε(t) ll

we

x Ma

Zener

igt

-Vo

in Kelv

t

Figure 2.4: Creep functions for three models

37

σ(t) r ne Ze

Kelvin-Voigt

Maxw

ell

t

Figure 2.5: Relaxation functions for three models

Figure 2.6: The mini-oscillators mechanical analogy in GHM method

38

7

10

Expt. GHM Fit

6

10

5

10

4

10

Storage Modulus

3

10

Loss Factors

2

10

1

10

0

10

−1

10

0

10

1

2

10

10

3

10

Frequency: Hz

Figure 2.7: The GHM prediction of complex shear modulus using three minioscillators

39

Chapter 3

Comparison of Analyses of Sandwich Beams

The structural system under consideration in this study is a three-layer sandwich beam which is comprised of two isotropic face layers sandwiching a viscoelastic core. Although this class of sandwich structures has been investigated extensively, it is examined here because of its simplicity and importance in understanding the fundamental physics of sandwich structures incorporating a viscoelastic core. First, we present the governing equations for sandwich beam and discuss the basic assumptions for this type of structures. Our contribution to sandwich beam analyses is to develop a spectral finite element method (SFEM) in the frequency domain. This SFEM provides an exact solution for sandwich beam because the shape functions are duplicated from progressive wave solutions. Because this method was developed in the frequency domain, there is no additional damping model needed and the frequency dependent complex shear modulus of

40

viscoelastic core can be accounted implicitly. We compare this method to general techniques of structural analyses, such as assumed modes (AM) and conventional finite element method (CFEM). The GHM method was applied to both analyses in order to account for the frequency dependent complex shear modulus of the viscoelastic core. Two examples of beams with passive constrained layer damping(PCLD) treatment were considered. One has 75% length of PCLD treatment as shown in Figure 3.1 and the other has 50% length of PCLD treatment as shown in Figure 3.2. Experiments were conducted to validate the analyses for those two beams. This chapter has been accepted for the publication by the ASME Journal of Vibration and Acoustics and in Reference [66].

3.1 3.1.1

Assumptions and Governing Equations Assumptions

Figure 3.3 shows the cross section of beam with PCLD treatment. The three displacements considered are the longitudinal displacements, u1 and u3 in the face layers 1 and 3, and the transverse displacement w for the whole sandwich beam. Mead [45] summarized the assumptions used in the modeling of beam with PCLD treatment. We modified it and included the longitudinal effects in the face layers. The assumptions are:

41

1. the viscoelastic core carries shear only and has a frequency dependent complex shear modulus; 2. the face layers are elastic and isotropic and suffer no transverse shear deformation; 3. the inertia of transverse and longitudinal effects in face layers are considered; the rotatory inertia of face layers are neglected and the viscoelastic core only contributes to the transverse inertia; 4. all points on the plate move with the same transverse displacement; 5. no slip occurs at the interfaces of the core and face layers. The above assumptions will be violated if the thickness of the viscoelastic core is of the same order as the base beam or constraining layer. In this case, the base beam and constraining layer will not have the same transverse displacement. A compression damping mechanism can occur in the viscoelastic core because of the relative transverse motion between base beam and constraining layer. This mechanism was discussed by Douglas [18]. In our studies, the viscoelastic core is very thin relative to isotropic face layers, with a maximum thickness of 10 mil. The shear damping is dominant in the viscoelastic core and covers a wide range of frequency for a thin sandwich beam, as shown in [18], so compression damping is negligible. Austin [1] discussed thickness effects in viscoelastic core, constraining layer and base beam and evaluated the assumptions for the different sandwich

42

beam configurations. Our assumptions are validated for our thin layer sandwich beam configurations. Based on above assumptions, the shear deformations in the viscoelastic core can be expressed in terms of displacements in face layers, as shown in Figure 3.4. The shear strain is: γ=

d ∂w (u1 − u3 ) + h2 ∂x h2

(3.1)

where d = h2 +

3.1.2

h1 + h3 2

(3.2)

Governing Equations

We can write the kinetic energy T and potential energy U of a sandwich beam: 1 T = 2

  2 2 
l   2 ∂w ∂u1 ∂u3 + m1 + m3 m dx ∂t ∂t ∂t 0

  2 2
l  1 ∂u1 ∂u3 E1 A1 U = + E3 A3 2 ∂x ∂x 0   2 2 ∂ w +Dt + GA2 γ 2 dx ∂x2

(3.3)

where m = m1 + m2 + m3

(3.4)

Dt = E1 I1 + E3 I3

(3.5)

The complex shear modulus, G⋆ , is composed of two components, the in-phase (real) part, G′ , and quadrature part (imaginary) part, G′′ . They are defined in

43

the frequency domain. But the total energy of a sandwich beam will be a complex number if we introduce the complex shear modulus G⋆ into the expression directly. This does not make sense physically. Therefore, we use the real component of shear modulus, G′ = G, in our derivation and replace it by the complex shear modulus after the governing equation of motion has been derived. More strictly, the complex shear modulus can only be used in the frequency domain for the forced response because it was developed under a sinusoidal force input. However, we still use it in the time domain as a simple representation. If we want to represent the complex shear modulus in the time domain, an additional damping model has to be introduced to capture the behavior of viscoelastic materials, for example, the GHM method. The equation of motion can be obtained by applying the Hamiltonian principle:
t2

δ(T − U)dt = 0

(3.6)

t1

The resulting equations of motion are as follows and the complex shear modulus, G⋆ , was introduced to replace the in-phase component.  ∂2w ∂4w G⋆ bd ∂u1 ∂u3 ∂2w − +d 2 m 2 + Dt 4 = ∂t ∂x h2 ∂x ∂x ∂x  2 ⋆ 2 ∂ u1 Gb ∂w ∂ u1 u1 − u3 + d −m1 2 + E1 A1 2 = ∂t ∂x h2 ∂x  2 ⋆ 2 ∂ u3 G b ∂w ∂ u3 −m3 2 + E3 A3 2 = − u1 − u3 + d ∂t ∂x h2 ∂x The associated boundary conditions on x = 0 and x = l are:

44

(3.7)

δu1 = 0

1 =0 or E1 A1 ∂u ∂x

δu3 = 0

3 or E3 A3 ∂u =0 ∂x

δw = 0

or

δ

 ∂w  ∂x

=0

G⋆ bdγ h2

3

− Dt ∂∂xw3 = 0

2

or Dt ∂∂xw2 = 0

If we neglect the longitudinal inertia effects in face layer 1 and 3, the equations reduce to the same forms as shown in Mead and Marcus [44].

3.2

Spectral Finite Element Method

In the CFEM, polynomial shape functions are used. In order to capture the exact dynamics, many elements are needed because of lower order approximation for displacement functions. CFEM can provide a standard matrix representation for structural problems. The elemental mass and stiffness matrices can be easily calculated for different types of elements. For a whole structural analyses, a simple assembling procedure is conducted to account for the boundary conditions, junctions, and load location. The progressive wave solution method can provide an exact solution for the dynamic problems in the frequency domain. A disadvantage of the progressive wave solution is that the matrix has to be reconstructed every time in order to adapt to the changes of boundary conditions, junctions, and loading. An idea was developed to combine the advantages of both CFEM method and progressive wave solution method. For a uniform rod and beam, we can directly

45

solve the governing equations of motion under harmonic excitation. The steady state solutions are found by solving the coefficients in the wave solution representations at each frequency. One further step is to develop the nodal degrees of freedom to solve for these wave coefficients in order to formulate a dynamic stiffness matrix. The shape functions used here were duplicated from progressive wave solutions. For one-dimensional isotropic rod and beam structures, Doyle [17] presented the dynamic stiffness matrix. SFEM is based on Fast Fourier Transformation (FFT) and Inverse Fast Fourier Transformation (IFFT). Doyle presented a detailed discussion for the FFT/IFFT based method in structural dynamic analyses. This methodology has been applied to solve a sandwich rod with active constrained layer damping treatment by considering longitudinal waves only [7]. We will extend it to analyze sandwich beams with two isotropic face layers and a viscoelastic core [62]. Because exact shape functions are duplicated from progressive wave solutions, only a few elements, as compared to CFEM methods, are needed to calculate the frequency response functions. Only as many elements are needed as there are junctions between substructures of different impedance. We will demonstrate the application of SFEM for sandwich beam analyses.

46

3.2.1

Isotropic Rod and Beam

The SFEM formulae for the isotropic rod and beam were given by Doyle [17]. Here we summarize the spectral finite element method for the isotropic rod and beam. For a rod, the kinetic and potential energy for longitudinal vibration are: 1 T = 2


l

m

0

1 U = 2


l



EA

0

∂u ∂t



2

∂u ∂x

dx

2

dx

(3.8)

where EA a longitudinal stiffness found by the product of Young’s modulus and area of cross section; m is the mass per unit length; u is the displacement along the axial x direction and l is the length of a rod element. The governing equation is EA

∂ 2 u(x, t) ∂ 2 u(x, t) = m ∂x2 ∂t2

(3.9)

Considering a steady state solution for the displacement u, it yields u(x, t) = uˆ(x, ω)ejωt

(3.10)

Substituting the above solution into Eq. 3.9, the final progressive wave solution of uˆ is uˆ(x, ω) = Ae−ikx + Beikx

(3.11)

where the wave number is given by k=ω



47

m EA

(3.12)

A and B are the unknown wave coefficients. Now a two node element is considered and the nodal displacement uˆ1 and uˆ2 are chosen to solve for the unknown A and B in the progressive wave solution expression. Then we can reconstruct the total energy expression in terms of nodal displacements of uˆ1 and uˆ2 in the frequency domain. It is ˆ = 1 {ˆ q }T kˆe {ˆ q }T Vˆ = Tˆ + U 2

(3.13)

where the nodal displacement vector qˆ is defined as: {ˆ q} = { uˆ1 uˆ2 }T kˆe is a 2 × 2 dynamic stiffness matrix which was given by Doyle [17]:   kˆe (ω) =

1 + e−i2kl −2e−ikl  EAik     1 − e−i2kl  −2e−ikl 1 + e−i2kl

(3.14)

(3.15)

We have a spectral finite element for rod longitudinal vibration. The same procedure as in CFEM is followed to assemble the elements for rod structures. A time domain force function is transformed to a frequency domain spectrum through the FFT transform and a time domain realization of a solution can be achieved by the IFFT transform. We can follow the same procedure to establish a dynamic stiffness matrix for an isotropic Bernoulli-Euler beam under transverse bending vibration. The governing equation is: ∂ 4 w(x, t) ∂ 2 w(x, t) EI +m =0 ∂x4 ∂t2

48

(3.16)

where EI is the flexural stiffness, m is the mass per unit length, and w is the displacement of transverse motion of a beam. The transverse displacement under harmonic excitation can be expressed as a wave expansion: w(x, ˆ ω) = Ae−ikx + Beikx + Ce−kx + Dekx

(3.17)

where the wave number is: mω 2 k= EI 

 14

(3.18)

Similar to the rod case, a two node element is chosen which is same as conventional beam finite element. The corresponding nodal degrees of freedom are two transverse displacements, w ˆ1 and wˆ2 and the slopes at two nodes,

∂w ˆ1 ∂x

and

∂w ˆ2 . ∂x

After mathematical manipulations, we can obtain a symmetric 4 × 4 dynamic ˆ stiffness matrix K. 

3 ik 3 k3 −k 3  −ik    k2 k2 −k 2 −k 2   ˆ K = EI   ik 3 e−ikl −ik 3 eikl −k 3 e−kl k 3 ekl    −k 2 e−ikl −k 2 eikl k 2 e−kl k 2 ekl −1 

     ×     

1

1

1

1

−ik

ik

−k

k

e

kl

e

−ikl

ikl

e

−kl

e

−ike−ikl ikeikl −ke−kl k1 ekl

          

           

(3.19)

This dynamic stiffness can be assembled to analyse the beam structure. The shape functions duplicated from progressive wave solutions provide an exact

49

solution for the displacement at each frequency.

3.2.2

Sandwich Beam

We have obtained the dynamic stiffness matrix for isotropic rod and beam structures. Now we will extend this methodology to sandwich beams with a viscoelastic core. For a spectral finite element method, we assume a harmonic motion at frequency ω. The general steady state solutions for displacement u1 , u3 and w can be expressed as: w(x, t) = w(x, ˆ ω)ejωt u1 (x, t) = uˆ1 (x, ω)ejωt u3 (x, t) = uˆ3 (x, ω)ejωt

(3.20)

By substituting the above expression into Eq. 3.7, we obtain governing equations in the frequency domain:  ∂ 4 wˆ G⋆ bd ∂ uˆ1 ∂ uˆ3 ∂ 2 wˆ − +d 2 −mω wˆ + Dt 4 = ∂x h2 ∂x ∂x ∂x  ⋆ 2 ∂ wˆ Gb ∂ uˆ1 2 uˆ1 − uˆ3 + d m1 ω uˆ1 + E1 A1 2 = ∂x h2 ∂x  ⋆ 2 ∂ wˆ G b ∂ uˆ3 2 uˆ1 − uˆ3 + d m3 ω uˆ3 + E3 A3 2 = − ∂x h2 ∂x 2

50

(3.21)

The solutions for uˆ1 , uˆ3 and wˆ can be expressed in terms of an expansion of waves: wˆ = W ekx uˆ1 = U1 ekx uˆ3 = U3 ekx

(3.22)

When substituting the above expressions into Eq. 3.21, the wave numbers k can be determined by assuming non-zero solutions for W , U1 and U3 . The characteristic equation is: θ 4 + λ3 θ 3 + λ2 θ 2 + λ1 θ + λ0 = 0 where G⋆ b G⋆ b G⋆ bd2 m1 ω 2 m3 ω 2 + − − − E1 A1 E3 A3 E1 A1 h2 E3 A3 h2 Dt h2  2 ⋆ 2 2 2 mω m1 m3 ω 4 G bd m1 ω m3 ω = − + − + Dt Dt h2 E1 A1 E3 A3 E1 A1 E3 A3 ⋆ G b (m1 ω 2 + m3 ω 2 ) − E1 A1 E3 A3 h2 mm1 ω 4 mm3 ω 4 G⋆ bd2 m1 m3 ω 4 = − − − Dt E1 A1 Dt E3 A3 Dt E1 A1 E3 A3 h2 mω 2 G⋆ b 1 1 + ( + ) Dt h2 E1 A1 E3 A3 mm1 m3 ω 6 mω 4 G⋆ b = − + (m1 + m3 ) Dt E1 A1 E3 A3 Dt E1 A1 E3 A3

λ3 = λ2

λ1

λ0 and

θ = k2

51

(3.23)

√ This equation provides four roots of θ1,2,3,4 and k = ± θ1 , etc. The solution results in eight values of k. Therefore, the general expression for w, ˆ uˆ1 and uˆ3 can be rewritten as: wˆ = uˆ1 =

4   i=1 4  i=1

uˆ3 =



4  

ai eki x + a−i e−ki x



bi eki x + b−i e−ki x



ci eki x + c−i e−ki x



i=1

(3.24)

When substituting the above equations into Eq. 3.21 and solving for uˆ1 and uˆ3 in terms of w, ˆ we will only have eight unknown wave coefficients ai , a−i (i = 1, 4). So bi , b−i and ci , c−i can be written as: bi = Yi ai b−i = −bi ci = µi bi c−i = −ci

i = 1, 4

(3.25)

where Yi µi

   h2  1 2 4 2 −mω + Dt ki − dki = (1 + µi )ki G⋆ bd m1 ω 2 + E1 A1 ki2 i = 1, 4 = − m3 ω 2 + E3 A3 ki2

(3.26) (3.27)

Again we have eight unknown wave coefficients ai , a−i (i = 1, 4). For the two node spectral finite element shown in Figure 3.5, the eight nodal displacements

52

are needed to solve the unknown wave coefficients. They are: {ˆ q} =



uˆ11 uˆ31 wˆ1

∂w ˆ1 ∂x

uˆ12 uˆ32 wˆ2

∂w ˆ2 ∂x

T

(3.28)

Now we can use nodal displacements to solve for wave coefficients of ai , a−i (i = 1, 4). Thus {ˆ q} = H{A}

(3.29)

where A is a vector of independent wave coefficients and is defined as: A=



a1 a2 a3 a4 a−1 a−2 a−3 a−4

T

(3.30)

H is a 8 × 8 transformation matrix which can be partitioned to H=



H1 H2

where the sub matrices H1 and H2 are: 

H1



Y1 Y2 Y3 Y4     µ Y µ2 Y2 µ3 Y3 µ4 Y4 1 1     1 1 1 1     k1 k2 k3 k4  =    Y ek1 l Y2 ek2 l Y3 ek3 l Y4 ek4 l  1     µ1 Y1 ek1 l µ2 Y2 ek2 l µ3 Y3 ek3 l µ4 Y4 ek4 l    ek2 l ek3 l ek4 l  ek1 l   k1 ek1 l k2 ek2 l k3 ek3 l k4 ek4 l

53

(3.31)

                           

(3.32)



H2

−Y1 −Y2 −Y3 −Y4     −µ Y −µ2 Y2 −µ3 Y3 −µ4 Y4 1 1     1 1 1 1     −k1 −k2 −k3 −k4  =    −Y e−k1 l −Y2 e−k2 l −Y3 e−k3 l −Y4 e−k4 l  1     −µ1 Y1 e−k1 l −µ2 Y2 e−k2 l −µ3 Y3 e−k3 l −µ4 Y4 e−k4 l    e−k1 l e−k2 l e−k3 l e−k4 l    −k1 e−k1 l −k2 e−k2 l −k3 e−k3 l −k4 e−k4 l

uˆ1 , uˆ3 and wˆ can be expressed in terms of the nodal displacements as



              (3.33)             

wˆ = {z}H −1 {q} = Nw {q} uˆ1 = {z}Yb H −1 {q} = Nu1 {q} uˆ3 = {z}Yc H −1 {q} = Nu3 {q}

(3.34)

The shape functions are defined as Nw = {z}H −1

(3.35)

Nu1 = {z}Yb H −1

(3.36)

Nu3 = {z}Yc H −1

(3.37)

54

where 

              =              

Yb

Y1

0

0

0

0

0

0

0

0

Y2

0

0

0

0

0

0

0

0

Y3

0

0

0

0

0

0

0

0

Y4

0

0

0

0

0

0

0

0

−Y1

0

0

0

0

0

0

0

0

−Y2

0

0

0

0

0

0

0

0

−Y3

0

0

0

0

0

0

0

0

−Y4

                            

(3.38)



Yc

0 0 0 0 0 0 0  µ1 Y1    µ2 Y2 0 0 0 0 0 0  0     0 0 µ3 Y3 0 0 0 0 0     0 0 0 µ4 Y4 0 0 0 0  =    0 0 0 0 −µ1 Y1 0 0 0     0 0 0 0 0 −µ2 Y2 0 0     0 0 0 0 0 0 −µ3 Y3 0    0 0 0 0 0 0 0 −µ4 Y4 {z} =



ek1 x ek2 x ek3 x ek4 x e−k1 x e−k2 x e−k3 x e−k4 x





               (3.39)              (3.40)

The spectral finite model of the sandwich beam can now be developed using the total spectral energy Eˆs of an element at length of l. ˆs = E

1 2


l 0

(E1 A1



dˆ u1 dx

2

+ E3 A3



dˆ u3 dx

2

+ Dt



d2 w ˆ dx2

+G⋆ A2 γˆ 2 − mω 2 w ˆ 2 − m1 ω 2 u ˆ21 − m3 ω 2 u ˆ23 )dx

55

2 (3.41)

Substituting the expression of displacement u ˆ1 , u ˆ3 and w ˆ as shown in Eq 3.34, yields ˆ e {ˆ ˆs = 1 {ˆ q T }K q} E 2

(3.42)

ˆ e is a dynamic stiffness of sandwich beam element. The expansion of the where K dynamic stiffness is: ˆe = K


l 

T

T

T

E1 A1 Nu′ 1 Nu′ 1 + E3 A3 Nu′ 3 Nu′ 3 + Dt Nw′′ Nw′′

0

+G⋆ A2 N T N − mω 2 NwT Nw  −m1 ω 2 NuT1 Nu1 − m3 ω 2 NuT3 Nu3 dx

(3.43)

where N=

Nu1 − Nu3 + dNw′ h2

(3.44)

and (.)′ denotes a derivative with respect to x. Compared to the previous dynamic stiffness matrix results for an isotropic rod and beam, we cannot obtain an explicit expression for the elements of the dynamic stiffness matrix in the sandwich beam case. However we can use numerical integration to obtain all the entries in the dynamics stiffness matrix. One special case is when the base beam and constraining layer are of the same material. Then in Eq. 3.27, the parameter µi will be reduced to: µi =

h1 h3

This constant ratio can cause a singularity in the matrix H. To remedy this case, we assume a perturbation of material density. In our calculation we assume that: m1 = ρbh1 m3 = (1 + ǫ)ρbh3

56

Usually ǫ = 0.0001 will give us a stable calculation of the numerical inverse of the matrix H. In the SFEM method, we do not need to use additional internal coordinates to model the viscoelastic core as shown in the GHM method. The frequency dependent complex shear modulus of the viscoelastic core were implicitly accounted in the frequency domain because the SFEM method was developed in the frequency domain. Also the interpolation functions were duplicated from a progressive wave solution. Then, the SFEM leads to a higher order method for the sandwich beam analysis.

3.3

Conventional Finite Element Method

Nostrand et al [49] provided a finite element model for the beam with active constrained layer damping (ACLD). The combinations of one-dimensional rod and beam finite elements were used to discretize the system. We followed a similar approach to study the dynamics of beam with PCLD [61]. A two-node element was used to approximate the displacement field for longitudinal displacements in face 1 and 3, u1 and u3 , and transverse displacement w. The nodal degrees of freedom are, as shown in Figure 3.6: {q} =



u11 u12 u31 u32 w1

∂w1 ∂x

w2

∂w2 ∂x

T

(3.45)

Therefore, in face layer 1 and 3, the longitudinal displacements u1 and u3 are:     u  11   u1 = 1 − xl xl    u12     u  31   x x u3 = (3.46) 1− l l    u32

57

And the transverse displacement w for the whole sandwich beam is:  w=



2

3

2

1 − 3 xl2 + 2 xl3

x − 2 xl +

2

x3 l2

3

2

3 xl2 − 2 xl3

− xl +



 w1        w′   1         w2      w2′

x3 l2

′

(3.47)

where l is the length of sandwich beam element, and (.) denote the derivatives respect to x. Substituting the above equations, Eq. 3.46 and Eq. 3.47 into Eq. 3.3, yields T

=

U

=

1 T {q }Me {q} 2 1 T {q }Ke {q} 2

(3.48)

and we can calculate the elemental mass matrix, Me , and stiffness matrix, Ke . Here we present the final results for both matrices. The stiffness matrix is:    k11 k12   ke =    k12 k22

where 

k11

EA1 l

G⋆ A 2 l 3h22

+     − EA1 + G⋆ A2 l  l 6h22 =   ⋆  − G3hA22 l  2   ⋆ − G6hA22 l

1 − EA l

EA1 l

−

k12

+

G⋆ A 2 l 3h22

G⋆ A

2l

⋆

    − G⋆ A 2 d  2h22 =    G⋆ A 2 d  2h22   G⋆ A 2 d 2h22

⋆ − G6hA22 l 2

⋆

− G3hA22 l

⋆

− G6hA22 l 2

+

2

G⋆ A

3 − EA l +

2

⋆ − G2hA22 d 2

⋆ − G3hA22 l 2

EA3 l

6h22

− G3hA22 l

2



+

G⋆ A 2 l 6h22

2l

3 − EA l +

3h22 G⋆ A 2 l 6h22

EA3 l

G⋆ A 2 d 12h22

G⋆ A 2 d 2h22

⋆A d 2 − G12h 2 2

A2 d − G12h 2

⋆

G⋆ A 2 d 2h22

G⋆ A 2 d 12h22

G⋆ A

⋆

12h22

− G2hA22 d

G⋆ A 2 d 12h22

G⋆ A 2 d 12h22

⋆ − G2hA22 d 2

A2 d − G12h 2

2

−

58

2d

(3.49)

2

⋆

2

           

+

G⋆ A 2 l 6h22

G⋆ A 2 l 3h22

           



k22

12Dt l3

6G⋆ A2 d2 5lh22

+     6Dt + G⋆ A2 d2  l2 10h22 =    12Dt 6G⋆ A2 d2  − l3 − 5lh2 2   G⋆ A2 d2 6Dt l2 + 10h2 2

6Dt l2 4Dt l

+ +

G⋆ A2 d2 10h22 2G⋆ A2 d2 l 15h22

t − 6D l2 −

2Dt l

−

The mass matrix is:

G⋆ A2 d2 10h22

G⋆ A2 d2 l 30h22



t − 12D l3

−

t − 6D − l2

12Dt l3

+

6G⋆ A2 d2 5lh22

6Dt l2

+

G⋆ A2 d2 10h22

G⋆ A2 d2 10h22

2Dt l

−

G⋆ A2 d2 l 30h22

6G⋆ A2 d2 5lh22

t − 6D l2 −

G⋆ A2 d2 10h22



 M11 04×4   Me =    04×4 M22

where



M11 =

0 0  2m1 m1    m 2m1 0 0 1 l  6  0 2m3 m3  0   0 0 m3 2m3

4Dt l

+

G⋆ A2 d2 10h22

2G⋆ A2 d2 l 15h22

          

(3.50)

            



M22 =

t − 6D l2 −



54 −13l   156 22l       22l 2 4l 13l −3l  ml      420   13l 156 −22l   54     −13l −3l −22l 4l2

where G⋆ is a complex shear modulus of the viscoelastic core. The GHM method has to be used to account for the frequency dependent properties of shear modulus as discussed in section 2.3.3.

3.4

Assumed Modes Method

The simplest analytical technique is the assumed modes method. The displacements u1 , u3 , and w are assumed as an expansion of the mode shapes functions which are

59

obtained from uniform rod longitudinal vibration and beam transverse bending vibration. w =

n 

Wi φiw

i=1

u1 =

m 

U1i ψui 1

i=1

u3 =

m 

U3i ψui 3

(3.51)

i=1

where n is total number of bending modes and m is total number of longitudinal vibration modes included. Those beam bending mode shapes and rod extension mode shapes are available in Chapter 6 of [27]. Substituting the above expressions for the displacement functions into the sandwich beam energy equation as shown in Eq. 3.3 and using Lagrange’s equation, we can obtain a second order differential equation M q¨ + (Ke + Kv )q = F

(3.52)

where ¯ Kv = G⋆ K where M is a mass matrix and Ke and Kv are stiffness matrices due to elasticity and damping respectively, F is a discretized forcing vector, and q is the known modal amplitudes as shown in Eq. 3.51. Details were presented in [61]. Similarly, the GHM method was used to account for the frequency dependent complex shear modulus of the viscoelastic core.

60

3.5

Solution Type/Methods

In this section, SFEM, CFEM and AM analyses are performed for the sandwich beam configurations shown in Figures 3.1 and 3.2. For each, the base beam (layer 1) is 508 mm (20 inches) long, 25.4 mm (1 inch) wide and 1.5875 mm (1/16 inch) thick. The constraining layer (layer 3) is 25.4 mm (1 inch) wide and 0.3969 mm (1/64 inch) thick. For the first specimen, the length of the PCLD treatment is 75% of the base beam length and for the second specimen, the length of the PCLD treatment is 50% of the base beam length. Cantilevered boundary conditions were simulated and both beams were actuated by a surface bonded PZT pair. The bending modal frequencies and tip frequency response functions were calculated. In the SFEM analysis, we use only four elements in the method to describe the dynamic response for the specimen 1. In the specimen 1 as illustrated in Figure 3.1, two isotropic beam elements are used for the sections of 0 < x < x1 and x2 < x < x3 and one element is used to model the section with piezo-actuators by modified stiffness and mass. For the beam with PCLD treatment, only one element is used to capture the dynamics. Similarly, in specimen 2, there are a total five elements used. The number of elements in our analysis coincides with the number of discontinuities of the beam with PCLD because the shape functions were duplicated from exact wave solutions. For CFEM method, 18 elements are used in the analysis for both specimens. Three elements are used for the section 0 < x < x3 , and for the section with PCLD treatment, x3 < x < x4 , fifteen elements are used, 1 inch long for each element. In the specimen

61

2, the exact same number of elements are used for section 0 < x < x3 . Ten elements are used for the sandwich part and another five elements are used for the isotropic beam from x4 < x < x5 . For specimen 2, the element mesh for SFEM and CFEM is shown in Figure 3.8. For the assumed modes method, the first twenty beam bending modes and first ten extension modes were used for both sandwich beams. The GHM method was used to account for the frequency dependent complex shear modulus of viscoelastic core in the CFEM and AM. model so that both analyses can be used to validate our SFEM results. Because PZTs were used to excite the sandwich beams, we have to model actuation force introduced in the sandwich beam system. A line moment is realized at the two edges of th PZT, as shown [10, 11, 13, 14]. The acutation moment was given as Mc = Ec bd31 V (h + hc )

(3.54)

where Ec is th Young’s modulus of the PZT, d31 is the piezo constant, V is the applied voltage, hc is the thickness of the piezo, and b is the width of the piezo which is same as the base beam, and h is the thickness of the base beam. Finally, the virtual work done by PZT is δW = −Mc δ



∂w ∂x



+ Mc δ

x=x2



∂w ∂x



(3.55) x=x3

The stiffness and mass of effects of PZT were considered by modified mass per unit ¯ for the element with PZT pair. They are: length, m, ¯ and bending stiffness, EI m ¯ = mb + mc  h2 2 2 Eb bh3 ¯ hc + hhc + + EI = Ec hc b 3 2 12

62

(3.56)

where mb is the mass per unit length of base beam and Eb is the Young’s modulus of the base beam. Therefore, a force vector can be calculated using the results of actuation moments. In the SFEM, the frequency response functions were calculated at each excitation frequency in which frequency dependent complex shear modulus is implicitly considered. The natural frequencies can be extracted from the response functions. In the CFEM and AM method, where both methods were augmented by the GHM method to account for frequency dependent complex shear modulus of the viscoelastic core, a state space model was used to represent the system to solve the eigenvalue problems for natural frequency, and Bode plots were calculated for frequency responses.

3.6

Experimental Setup

The analyses have been validated experimentally using frequency response data measured from cantilevered aluminum beams with passive constrained layer damping (PCLD), as shown in Figure 3.7. The Siglab data acquisition system was used to generate input and collect output through a computer. The sinusoidal signal which was amplified by a power amplifier was applied to the PZT to excite the beam. A Schaevitz DistanceStar laser sensor was used to measure the tip response under the sine sweep signal. The material constants for the beams are listed in Table 3.1. The viscoelastic damping material is 3M Scotchdamp ISD 112, and is 0.127 mm (5 mil) thick for our setup. Both the storage modulus and loss factors are dependent on frequency and temperature, based on data provided by 3M [56]. The GHM model

63

included three mini-oscillators in the expansion of the materials properties and the constants were shown in Section 2.3.3.

3.7 3.7.1

Results Modal Frequency Predictions

The modal frequencies predicted from all analyses of sandwich beams using the SFEM, CFEM and AM method are presented. All analyses are performed for the sandwich beam configurations shown in Figures 3.1 and 3.2. Table 3.2 and 3.3 show a comparison between the predicted and measured modal frequencies for specimen 1 (75% PCLD) and specimen 2 (50% PCLD) respectively. We can see in these tables that SFEM provides more accurate modal frequency predictions than those in the CFEM and AM. SFEM uses only 4 elements for specimen 1, for a total of 14 degrees of freedom, and 5 elements for specimen 2, for a total 17 degrees of freedom. On the other hand, the CFEM analyses used 18 elements, was augmented further with internal dissipation coordinates to account for the frequency dependent complex modulus, for a total of 280 degrees of freedom in specimen 1, and for a total of 260 degrees of freedom for specimen 2. In the AM method, the first twenty beam bending mode shapes and first ten extension mode shapes were used to obtain comparable results. As with the CFEM, AM analysis used the GHM method to account for the frequency dependent complex shear modulus of the viscoelastic core. In the SFEM method, the prediction errors for the first through fifth modal frequencies are smaller than those in the CFEM and AM prediction. The spectral finite element method provides more accurate predictions of

64

higher modal frequency, while using only a small fraction of the number of elements used by CFEM. Table 3.3 shows the modal frequency comparisons for specimen 2 (50% PCLD), where an additional structural junction is added at x4 = 0.75L. Compared to the results of specimen 1, the prediction error is much higher for the CFEM and AM analyses, whereas the SFEM predictions remain comparable to those of specimen 1. An additional impedance change or discontinuity in the structure is more easily handled by the SFEM method by simply introducing an additional spectral finite element to handle the new discontinuity whereas the CFEM and AM predictions degrade substantially. The CFEM and AM analyses can provide modal frequency predictions for a beam with PCLD treatment; however, more elements and more modes must be included to achieve comparable results with respect to experimental data. The SFEM is a higher order method, so it can provide a better prediction with less computational cost because only fewer elements were employed.

3.7.2

Number of Elements

In Figure 3.9, we investigate the typical impact of the number of elements on accuracy of modal frequency predictions by the CFEM and SFEM analyses for specimen 1 (75% PCLD). We plot the number of elements versus the non-dimensional modal frequencies with respect to experimental results for the first five modes. Again, the CFEM analysis uses GHM to account for the viscoelastic core. The shape functions in CFEM are the polynomial functions intended to interpolate the displacement across each element.

65

Typically, the number of elements must be substantially increased in order to increase the accuracy of modal frequency predictions. As shown in Figure 3.9, increasing the number of elements in the CFEM analysis improves accuracy. In addition, as the number of elements increases, higher modal parameters can be predicted with more confidence. For example, for the case of 6 or 8 element number, the error for higher modal frequencies (from 2 to 5 mode) is large. When we increase the number of elements to 33 elements, we can match experimental results. Increasing the number of elements has a smaller impact on reducing prediction errors in the lower modes. The typical analysis perspective is to match the first modal frequency by adjusting material parameters. As shown in Figure 3.9, the same number of elements were used for SFEM analysis. A key aspect of the SFEM analysis for this structure is that increasing the number of elements does not improve the prediction errors. In fact, the modal frequency prediction error for N = 4 elements is identical to that for N = 33 elements. The refined shape functions, or exponential wave functions, increases the order of the approximate interpolation of displacement in our analysis so that the number of elements has no effect on the results. We need only include the number of elements that correspond with the discontinuities or junctions of the beam with PCLD. We can obtain similar results for specimen 2 (50% PCLD) as shown in Figure 3.10.

3.7.3

Frequency Response Functions

Figures 3.11 and 3.12 show the frequency response functions (FRFs) from 1 Hz to 400 Hz for the beam specimen with 75% and 50% PCLD treatments, respectively. In these

66

calculations, the SFEM analysis for specimen 1 (75% PCLD) used N = 4 elements and specimen 2 (50% PCLD) used N = 5 elements. The CFEM used 18 elements for both specimens. The first twenty beam bending modes and first ten rod modes are included in AM method. The analytical methods capture the trend of the FRFs in both magnitude and phase. The SFEM analysis proved to be more accurate for high frequency even though only 4 or 5 elements were used in the analysis.

3.8

Summary

We present a spectral finite element model (SFEM) for sandwich beams excited with a pair of piezoelectric actuators. The sandwich beam consists of top and bottom aluminum face layers sandwiching a viscoelastic core. The viscoelastic core has a complex modulus that varies with frequency. The actuators are mounted on the top and bottom of the beam and are excited with equal, but out-of-phase, voltages to excite bending motion of the sandwich beam. The SFEM is formulated in the frequency domain using dynamic shape functions based on the exact displacement solutions from wave propagation methods where we implicitly account for the frequency dependent complex modulus of the viscoelastic core. A small number of spectral elements is required to calculate the frequency response functions of the sandwich beam. Existing analysis methods, the conventional finite element (CFEM) and assumed modes method (AM) were compared to the SFEM. Each of the analyses was compared to experimental measurements of modal frequency and frequency response functions for two specimen, the first having 75% PCLD treatment and the second having 50% PCLD treatment.

67

The SFEM method uses wave propagation functions that are exponential in nature to construct the displacement of the nodes for an element. The wave propagation functions used in SFEM are much higher order than the low order polynomial functions typically used in conventional FEM interpolate displacements from node to node in an element. The resulting dynamic stiffness matrix or impedance matrix is a function of frequency. The primary advantages of the SFEM method, described below, were demonstrated for these sandwich beams.

1. To account for the frequency dependent complex modulus of the viscoelastic core, internal dissipation coordinates are typically added to CFEM and AM analyses using either the GHM or ADF methods. Applying these internal dissipation coordinate methods substantially increases the required number of degrees of freedom in the CFEM and AM analyses. SFEM implicitly accounts for the frequency dependent complex modulus of the viscoelastic core without adding internal dissipation coordinates. This is because the dynamic stiffness or impedance matrix is computed at each frequency, so that the appropriate value of complex modulus is used at each frequency. 2. Substantially fewer elements were required by the SFEM analysis (N = 4 for the specimen with 75 % PCLD or N = 5 for the specimen with 50% PCLD). The number of degrees of freedom for CFEM (280/260 for specimen 1 and 2, respectively) analysis was substantially larger. Even though the CFEM used more elements, the SFEM substantially out-performed the CFEM in terms of modal frequency prediction accuracy when compared to experimental measurements of

68

the modal frequencies. 3. Increasing the number of spectral finite elements did not improve the modal frequency prediction accuracy for these simple (section-wise uniform) sandwich beams. The interpolation functions used are waves, and no benefit is derived from adding additional elements as long as the impedance does not change in the structure. Physically, this implies that as long as a section of the beam has uniform properties (as in our sandwich beams) regardless of its geometry, one element suffices to capture the structural response. In contrast, CFEM requires many elements to interpolate the bending displacement correctly. As a result, more conventional finite elements must be added as the beam length increases, in contrast to SFEM, which still would require only a single spectral finite element. For non-uniform structures, for example, a tapered beam with PCLD, the advantage in number of elements would be lost because the impedance changes continuously. The advantages and disadvantages of SFEM with respect to CFEM for non-uniform structures deserves further study. 4. SFEM provided better results that the CFEM analyses in predictions of the frequency response function (FRF) of the sandwich beams. In the FRF calculation, we can directly account for the frequency dependent complex shear modulus in the frequency domain. No constitutive relationship for the viscoelastic material is required beyond measurements of the complex modulus. SFEM improved its prediction of the FRF as frequency increased.

Because the dynamic stiffness matrix is calculated at each frequency, a parallel

69

algorithm can easily be applied to obtain the results simultaneously. This will lead SFEM to a faster execution. Although this study examined only passive constrained layer damping treatments, the SFEM analysis can be easily extended to active constrained layer damping treatments.

70

Table 3.1: Beam and actuator constants Piezoelectric density (kg/m3 )

7500

Piezoelectric Young’s modulus (GPa) 71 Piezoelectric constant (m/v)

-274e-12

Aluminum Young’s modulus (GPa)

69

Aluminum density (kg/m3 )

2700

Viscoelastic density (kg/m3 )

1000

Table 3.2: Comparison of predicted and measured modal frequencies of the beam specimen 1 with a 75% PCLD treatment. N is the number of finite elements; Nb is the number of bending modes and Ne is the number of extension modes used in AM method.

SFEM

CFEM

AM

N =4

N = 18

Nb = 20, Ne = 10

Modes

Expt

Anal

Error

Anal

Error

Anal

Error

No.

[Hz]

[Hz]

[%]

[Hz]

[%]

[Hz]

[%]

1

5.0

5.16

3.2

5.2

4.0

5.3

6.0

2

35.9

37.0

3.1

37.2

3.6

37.8

5.3

3

95.7

98.1

2.5

98.5

3.0

100.3

4.8

4

187.4 192.1

2.5

193.8

3.4

195.9

4.4

5

306.0 314.5

2.8

320.0

4.6

325.6

6.4

71

Table 3.3: Comparison of predicted and measured modal frequencies for beam specimen 2 with a 50% PCLD treatment. N is the number of finite elements; Nb is the number of bending modes and Ne is the number of extension modes used in AM method. SFEM

CFEM

AM

N =4

N = 18

Nb = 20, Ne = 10

Modes

Expt

Anal

Error

Anal

Error

Anal

Error

No.

[Hz]

[Hz]

[%]

[Hz]

[%]

[Hz]

[%]

1

5.3

5.5

3.8

5.6

5.7

5.7

7.5

2

35.1

36.5

4.0

36.7

4.6

37.5

6.8

3

87.0

90.2

3.7

90.5

4.0

91.7

5.4

4

174.5 177.1

1.5

178.2

2.1

180.7

3.5

5

292.5 301.6

3.1

304.0

4.0

310.2

6.1

72

x1

x4

x3 x2

Piezoelectric Actuator (PZT-5H)

Base Beam (Aluminum) Viscoelastic Damping Layer Constrained Layer (Aluminum)

Figure 3.1: Specimen 1: the PCLD treatment covers 75% of the total length of the beam

x5 x4

x1

x3 x2 Piezoelectric Actuator (PZT-5H)

Base Beam (Aluminum) Viscoelastic Damping Layer Constrained Layer (Aluminum)

Figure 3.2: Specimen 2: the PCLD treatment covers 50% of the total length of the beam

73

u1 h1

w

h2 h3 u3

Figure 3.3: Cross section of beam with PCLD treatment

z, w u1 w x

γ

u3 w

x,u

Figure 3.4: Deflection of beam with PCLD treatment

wˆ 1

ˆ1 ∂w ∂x

ˆ wˆ 2 ∂w2 ∂x

uˆ 11

uˆ 12

uˆ 31

uˆ 32

Figure 3.5: Nodal degrees of freedom in SFEM

74

w1

w2 w2 '

w1' u11

u12

u 31

u 32

Figure 3.6: Nodal degrees of freedom in CFEM

Power Amplifiers

SigLab Box

Laser sensor

+ _ Computer

Figure 3.7: Experimental set up for beam with PCLD treatments

75

SFEM 1 2

3

4

5

PCLD Treatment

CFEM 1 2

3

4-13

13-18

PCLD Treatment Figure 3.8: Number of elements used in SFEM and CFEM for 50% PCLD beam

1.05 Mode 1 1.04 1.03

5

10

15

20

25

30

35

25

30

35

25

30

35

25

30

1.06

Non−dimensional Modal Frequency

Mode 2 1.04 1.02

5

10

15

20

1.05 Mode 3 1.025 1

5

10

15

20

1.15 Mode 4

1.1 1.05 1

5

10

15

20

35

1.15 CFEM SFEM

Mode 5

1.1 1.05 1

5

10

15

20 Number of Elements: N

25

30

35

Figure 3.9: The effects of number of elements on modal frequencies for specimen 1 having 75% PCLD treatment

76

1.1 Mode 1 1.05 1

5

10

15

20

25

30

35

25

30

35

25

30

35

25

30

1.08

Non−dimensional Modal Frequency

Mode 2 1.04 1

5

10

15

20

1.06 Mode 3 1.04 1.02

5

10

15

20

1.1 Mode 4 1.05 1

5

10

15

20

35

1.1 CFEM SFEM

Mode 5 1.05 1

5

10

15

20 Number of Elements: N

25

30

35

Figure 3.10: The effects of number of elements on modal frequencies for specimen 2 having 50% PCLD treatment

77

−20

Mag. dB

−60 Expt. SFEM CFEM A.M

−100

−140 0 10

1

2

10

10 Frequency [Hz]

200

Phase [deg]

150

100

50

0 0

10

1

2

10

10 Frequency [Hz]

Figure 3.11: Frequency Response function from the piezoelectric voltage input to the tip displacement output: the PCLD treatment covers 75% of the length of the base beam.

78

−20

Mag. dB

−60 Expt. SFEM CFEM A.M

−100

−140 0 10

1

2

10

10 Frequency [Hz]

200

Phase [deg]

150

100

50

0 0

10

1

2

10

10 Frequency [Hz]

Figure 3.12: Frequency Response function from the piezoelectric voltage input to the tip displacement output: the PCLD treatment covers 50% of the length of the base beam.

79

Chapter 4

Analyses of Sandwich Plates: Part I

This chapter discusses the bending vibration of a plate with passive constrained layer damping treatment (PCLD). This 3-layer plate structure is comprised of two face plates and a viscoelastic core. The face plates are assumed to be isotropic materials and the viscoelastic core is a material with a frequency dependent complex shear modulus. The set-up of the problem is similar as the sandwich beam case discussed in chapter 3. We have demonstrated that a SFEM method which was developed in the frequency domain was used to calculate the response of a sandwich beam. This method provides the exact solutions for the vibration of sandwich beams and we expect to extend this method to sandwich plate analysis. It is very difficult to directly solve the problem of vibration of a sandwich plate that is rectangular. For a rectangular isotropic plate, there are no closed-form solutions for bending vibration except for the Levy type of plates, in which at least two parallel edges are simply-supported. Usually the Ritz method is used to calculate the natural frequency and response for two-dimensional plate structures. The assumed mode shapes used in the Ritz method

80

were approximated by the beam bending mode shapes for both x and y direction. We use this classical assumed modes method for the analysis of sandwich plate using beam and rod mode shapes, in which the GHM method was incorporated to account for the frequency dependent complex shear modulus. The current available experimental test data of three fully treatment of PCLD plates were used to validate the predictions of natural frequencies and loss factor. In this analysis, we need to include many mode shapes in our calculations, especially the in-plane mode shapes approximated by rod vibration mode shape functions in order to match experimental data. This chapter is an expanded version of Reference [64].

4.1

Assumptions and Governing Equations

4.1.1

Asssumptions

The configuration of 3-layered sandwich plate is illustrated in Figure 4.1. Layers 1 and 3 are the isotropic face-plates, made from aluminum, and the core is the viscoelastic material. The face-plates are assumed to have bending, in-plane shear and extensional stiffness and their rotatory inertia has also been neglected in the model. The viscoelastic core is assumed to have transverse shear stiffness alone. The assumptions involved in the derivation of the governing equations of a sandwich plate are: a. the face plates are elastic and isotropic and suffer no transverse shear deformation, that is, the Kirchhoff hypothesis; b. the core carries transverse shear, but no in-plane stresses; it is linearly viscoelastic

81

∂w ∂x ∂w ∂x

γ xz

γ yz

∂w ∂y ∂w ∂y

Figure 4.1: (a) Sandwich plate showing its co-ordinate axes and dimensions, and (b) Layers forming the sandwich, and the displacements associated with each layer. and has a complex modulus; c. no slip occurs in the interfaces between the face-plates and the core and all points normal to the plate move with the same transverse displacement.

The above assumptions are similar to sandwich beam analyses in chapter 3. In this case, we extend one-dimensional beam structures to two-dimensional plate structures. The shear strain in the viscoelastic core (layer 2) can be expressed by: u32 − u12 ∂w u3 − u1 d ∂w + = + h2 ∂x h2 h2 ∂x v32 − v12 ∂w v3 − v1 d ∂w = + = + h2 ∂y h2 h2 ∂y

γx,2 = γy,2

82

(4.1)

where d is the distance between the mid-plane of layer 1 and mid-plane of layer 3 and is defined as: d = h2 +

h1 + h3 2

(4.2)

Applying the assumptions, we can derive the strains in face layers in terms of face layer in-plane displacement ui (x, y) and vi (x, y), i = 1, 3 and transverse displacement w(x, y). They are ǫx,i = ǫy,i = ǫxy,i =

∂2w ∂ui −z 2 ∂x ∂x ∂2w ∂vi −z 2 ∂y ∂y ∂ui ∂vi ∂2w + − 2z 2 ∂y ∂x ∂x

(4.3)

and the stresses in face layers 1 and 3 are: σx,i = σy,i = σxy,i =

Ei (ǫx,i + νǫy,i ) 1 − ν2 Ei (ǫy,i + νǫx,i ) 1 − ν2 Ei ǫxy,i 2(1 + ν)

(4.4)

where Ei is the Young’s modulus for face layers, i = 1, 3 and we assume the Poisson ratio ν are same for both face layers.

4.1.2

Sandwich Plate Energies and Governing Equations

Similarly, we write down the total system energy for the sandwich plate and derive the governing equations of motion by applying the Hamiltonian principle. As discussed in Section 3.1, the complex shear modulus was not used in the derivation because the complex energy has no physical meaning. We replace the in-phase component by

83

the complex modulus later in the final governing equations. The kinetic energy of sandwich plate is: T

1 2

=




A



ρh



∂w ∂t

2

+



ρi hi

i=1,3



∂ui ∂t

2

+



 2  ∂ui  dA ∂t

(4.5)

where ρi is the density of each layers including viscoelastic core and hi is the thickness of the each layers The mass per unit area for a total sandwich plate is defined as: ρh = ρ1 h1 + ρ2 h2 + ρ3 h3

(4.6)

The total potential energy of sandwich plate, including the transverse bending, inplane energy in the face layers 1 and 3, and the transverse shear energy alone in the core, is: U

=

1 2




A

 



i=1,3

   ∂ui ∂vi ∂ui ∂vi Nx,i + Ny,i + Nxy,i + ∂x ∂y ∂y ∂x i=1,3

i=1,3

  ∂2w ∂2w ∂2w − M − 2 M y,i xy,i ∂x2 ∂y 2 ∂x∂y i=1,3 i=1,3 i=1,3  ¯ x,2 γx,2 + Q ¯ y,2 γy,2 dA +Q −



Mx,i

(4.7)

In layer 1 and 3, the resultant in-plane stresses are

Nx,i =

Ny,i =

h
i /2



∂ui ∂vi +ν ∂x ∂y

−hi /2

Ei hi 1 − ν2



h
i /2

Ei hi σy,i dz = 1 − ν2



∂vi ∂ui +ν ∂y ∂x



−hi /2

Nxy,i =

h
i /2

σx,i dz =

Ei hi σxy,i dz = 2(1 + ν)

−hi /2

84



∂ui ∂vi + ∂y ∂x



i = 1, 3

(4.8)

and bending moments are defined as: h
i /2



∂2w ∂2w + ν ∂x2 ∂y 2

−hi /2

Ei h3i σx,i zdz = − 12(1 − ν 2 )



h
i /2

Ei h3i σy,i zdz = − 12(1 − ν 2 )



∂2w ∂2w + ν ∂y 2 ∂x2



h
i /2

Ei h3i σxy,i zdz = − 12(1 + ν)



∂2w ∂x∂y

Mx,i =

My,i =

−hi /2

Mxy,i =

−hi /2



i = 1, 3

(4.9)

and shear stresses in viscoelastic core are h
2 /2

σxz,2dz = Gh2

d ∂w u3 − u1 + h2 h2 ∂x

−h2 /2



h
2 /2

σyz,2 dz = Gh2



d ∂w v3 − v1 + h2 h2 ∂y

¯ x,2 = G Q

¯ y,2 = G Q

−h2 /2





(4.10)

The symbol G will be replaced by the complex shear modulus G⋆ in the governing equations later. The variation of the kinetic energy is expressed by:   
 ∂ui ∂δui ∂vi ∂δvi  ρh ∂w ∂δw + δT = + dA ρi hi ∂t ∂t ∂t ∂t ∂t ∂t

(4.11)

i=1,3

A

The variation of total potential energy is:  
   ∂δvi ∂δui ∂δvi ∂δui  + Ny,i + Nxy,i + δU = Nx,i ∂x ∂y ∂y ∂x A

i=1,3

i=1,3

i=1,3

  ∂ 2 δw ∂ 2 δw ∂ 2 δw − My,i −2 Mxy,i 2 2 ∂x ∂y ∂xy i=1,3 i=1,3 i=1,3  1 ∂δw +Qx,2 δu3 − δu1 + d h2 ∂x   ∂δw 1 δv3 − δv1 + d + Qy,2 dA h2 ∂y −



Mx,i

(4.12)

The equation of motion can be obtained by applying the Hamiltonian principle:
t2

t1

δ(T − U )dt = 0

85

(4.13)

There are five partial differential equations corresponding to five independent displacements. Now we introduce the complex shear modulus of viscoelastic core G⋆ to replace the nominal real value G. We present the equations in terms of five displacements, inplane displacement in face layer 1, u1 (x, y) and v1 (x, y); in-plane displacement in face layer 3, u3 (x, y) and v3 (x, y); and transverse displacement w(x, y) for whole sandwich plate.   4 ∂4w ∂4w d ∂Qy,2 ∂ w d ∂Qx,2 ∂2w +2 2 2 + 4 + = ρh 2 + Dt 4 ∂t ∂x ∂x ∂y ∂ y h2 ∂x h2 ∂y

(4.14)

  Qx,2 ∂ 2 u1 1 ∂ 2 v1 Eh1 ∂ 2 u1 1 ∂ 2 u1 + (1 − µ) 2 + (1 + µ) = ρ1 h1 2 − 2 2 ∂t 1−µ ∂x 2 ∂y 2 ∂x∂y h2

(4.15)

  Eh1 ∂ 2 v1 1 ∂ 2 v1 1 ∂ 2 u1 Qy,2 ∂ 2 v1 + (1 − µ) 2 + (1 + µ) = ρ1 h1 2 − 2 2 ∂t 1−µ ∂y 2 ∂x 2 ∂x∂y h2

(4.16)

  Qx,2 ∂ 2 u3 ∂ 2 u3 1 ∂ 2 v3 Eh3 ∂ 2 u3 1 (1 − µ) (1 + µ) − + + =− ∂t2 1 − µ2 ∂x2 2 ∂y 2 2 ∂x∂y h2

(4.17)

  Eh3 ∂ 2 v3 1 ∂ 2 v3 1 ∂ 2 u3 Qy,2 ∂ 2 v3 + (1 − µ) 2 + (1 + µ) =− ρ3 h3 2 − 2 2 ∂t 1−µ ∂y 2 ∂x 2 ∂x∂y h2

(4.18)

ρ3 h3

Here the total bending flexural stiffness and shear force in the viscoelastic core are: E1 h1 E3 h3 + 2 12(1 − ν ) 12(1 − ν 2 )   d ∂w ⋆ = G u3 − u1 + h2 ∂x   d ∂w ⋆ = G v3 − v1 + h2 ∂y

Dt = Qx,2 Qy,2

(4.19)

The governing equations Eqs. 4.14 to 4.18 are associated with following possible boundary conditions along the four edges of a rectangular sandwich plate, which were also obtained from the Hamiltonian principle: (a) at x = 0, L.

86

δui = 0

or Nx,i = 0

δvi = 0

or Nxy,i = 0

δw = 0

or Qx = 0

δ∂w ∂x

=0

or Mx = 0

δvi = 0

or Ny,i = 0

δui

or Nxy,i = 0

δw = 0

or Qy = 0

δ∂w ∂y

or My = 0

and (b) at y = 0, C

=0

for i = 1, 3. where Mx = Mx,1 + Mx,3 My = My,1 + My,3 Mxy = Mxy,1 + Mxy,3 Qx = Qy =

∂Mx ∂Mxy d +2 + Qx,2 ∂x ∂y h2 ∂My ∂Mxy d +2 + Qy,2 ∂y ∂x h2

(4.20)

It is very difficult to directly solve the boundary values problem of PDEs. A numerical method is applied to obtain the results of natural frequency and loss factor in order to evaluate sandwich plate structures. The assumed modes method was employed to discretize the system and generates the stiffness and mass matrices. The GHM method is then applied to account for the frequency dependent properties of the complex shear modulus for the viscoelastic core. The predictions of natural frequency and loss factor were validated by the closed-form solutions for a simply-supported rectangular

87

sandwich plate and also the experimental data of three sandwich plates clamped on all edges.

4.2

Assumed Modes Method Using Beam and Rod modes

In order to calculate the natural frequencies and loss factor, the classical assumed modes method was used to analyze the sandwich plate with a viscoelastic core. The displacements were assumed as an expansion of mode shape functions with unknown weighted amplitudes. The five displacements for sandwich plate were assumed as: w(x, y, t) =



Wi (t)Φiw (x, y)

i

u1 (x, y, t) =



Ui1 (t)Ψiu1 (x, y)

i

v1 (x, y, t) =



V1i (t)Ψiv1 (x, y)

i

u3 (x, y, t) =



U3i (t)Ψiu3 (x, y)

i

v3 (x, y, t) =

 i

88

V3i (t)Ψiv3 (x, y)

(4.21)

where the ith mode shape function Φiw , Ψiu1 , Ψiv1 , Ψiu3 , and Ψiu3 were approximated by the beam and rod modes in both x and y direction. They are expressed as follows: n Φiw (x, y) = φm w (x)φw (y)

Ψiu1 (x, y) = ψum1 (x)φnu1 (y) Ψiv1 (x, y) = ψvm1 (x)φnv1 (y) Ψiu3 (x, y) = ψum3 (x)φnu3 (y) Ψiv3 (x, y) = ψvm3 (x)φnv3 (y)

(4.22)

For the ith assumed mode shape function, there is a corresponding mapping of mode numbers m and n in x and y direction, respectively. Table 4.2 shows the mapping relationship between the assumed modes number and corresponding modes in both x and y directions. Substituting the assumed mode shape functions for all the displacements as illustrated in Eq. 4.22 into the total energy expression in Eqs. 4.5 and 4.7 and applying Lagarange Equation, we can obtain the discretized second order ordinary differential equations as shown below: ¨ + Ke x + G⋆ Kv x = F Mx

(4.23)

The mass matrix, M , and stiffness matrices, Ke and Kv , are shown in Appendix. We introduce the GHM method to account for the frequency dependent complex shear modulus, G⋆ . where details of the method were discussed in Section 2.3.3. Finally, the state space model can be achieved based on the modified mass, damping, and stiffness as shown in Eqs. 2.32 to 2.34 based on the GHM method. The following sections

89

will demonstrate the assumed modes method for the sandwich analyses. The available analytical solutions for a simply supported sandwich plate and the experimental data of three four-side clamped sandwich plates were used to validate our results.

4.2.1

Analytical Validation: Simply Supported

This section compares the modal frequencies of free vibration predicted by an existing analytical solution [28] to those predicted by our analysis for a simply-supported sandwich plate with aluminum face-plates and a viscoelastic core. The complex shear modulus of the core is assumed constant over the frequency range, it is unnecessary to use the GHM method to account for the frequency dependent complex modulus of the viscoelastic core. For a simply-supported sandwich plate, the plate bending modes in w are of the form: w(x, y, t) =



Wi Φi (x, y) =

i



Wi (t) sin

i

nπy mπx sin L C

(4.24)

where L and C are the dimension of plate in x and y direction as illustrated in Figure 4.1. Thus, the assumed plate bending mode used is the product of the appropriate Euler-Bernoulli beam bending mode in each x and y direction. The assumed in-plane modes for ui and vi in the face plates 1 and 3 are the appropriate and mode shapes are of the form: mπx nπy sin L C i i   mπx nπy v1 (x, y, t) = V1i (t)Ψiv1 (x, y) = V1i (t) sin cos L C i i   nπy mπx sin u3 (x, y, t) = U3i (t)Ψiu3 (x, y) = U3i (t) cos L C i i   nπy mπx cos v3 (x, y, t) = V3i (t)Ψiv3 (x, y) = V3i (t) sin L C

u1 (x, y, t) =



U1i (t)Ψiu1 (x, y) =

i



i

90

U1i (t) cos

(4.25)

Here, m and n are obtained for the ith mode via the mapping in Table 4.2. Φ(x, y) and Ψ(x, y) are the plate bending and in-plane mode shape functions which were approximated by beam and rod modes. The boundary conditions for a simply-supported sandwich plate are: (a) at x = 0, L. Nx,1 , Nx,3 , v1 , v3 , w, Mx = 0

(4.26)

Ny,1 , Ny,3 , u1 , u3 , w, My = 0

(4.27)

and (b) at y = 0, C

where Nx,i , Ny,i , i = 1, 3 were defined in Eq. 4.8, and bending moments Mx and My were defined in Eq. 4.9. When we substitute the assumed mode shape functions as shown in Eqs. 4.24 and 4.25, all the boundary conditions were satisfied. This will give us a better predictions of natural frequency and loss factor. This is the reason that there are only closed form solutions available for a simply-supported sandwich plate with constant complex shear modulus. For the assumed mode method, the assumed modes need only satisfy all the geometric boundary conditions; for the simplysupported case we can find the exact mode shape functions. The terms W i , U1i , U3i , V1i and V3i are the coefficients of the corresponding natural mode shapes. The mass and stiffness matrices are obtained using the first 16 transverse bending and in-plane mode shapes. The sandwich plate was symmetric with isotropic face layers, where the length in the x direction is L = 0.348m and the length in the y direction is C = 0.3048m. The thickness of the each layer is that h1 = h3 = 0.762mm and h2 = 0.254mm. The material constant are shown in Table 4.1. The predicted modal frequencies and the corresponding modal loss factors are tabulated against the exact solution in

91

Table 4.1: Material constants for a simply-supported sandwich plate E1 , E3

68.9GP a

ρ1 , ρ3

2740kg/m3

ρ2

999kg/m3

G2

0.869MP a

η2

0.5

Table 4.3. The predicted values match the closed form analytical solution very well. The error is less than 1.2% for both the modal frequency, and loss factors. Our results also agree well with the numerical solutions of Cupial and Niziol [15].

4.2.2

Experimental Validation: All Four Sides Clamped

4.2.2.1

Set-up

This section presents an experimental validation of sandwich plates with aluminum isotropic face plates and viscoelastic cores. The details of this experimental set-up was presented by Veeramani [60] in her M.S thesis. We outline some key features. The test area of the plates is 67.31 cm x 52.07 cm (26.5” x 20.5”). The plates were clamped atop an aluminum stand using bars of cross section, 3.81 cm (1.5”) wide and 0.953 cm (3/8”) thick, around the perimeter, as shown in Figure 4.2. Each bolt was inserted through the top clamping bar, the plate, and the test stand, and then fastened with a nut. Each bolt was subsequently tightened to a constant torque as measured by a torque wrench. Piezo-actuators were fastened to the plate, and used to excite bending

92

motion of the plate and the response of the plate was measured using accelerometers. The stand was calibrated by testing uniform aluminum plates of varying thickness. Table 4.5 shows the results of calibration. For a thickness of 0.16 cm (1/16”) the error in the first modal natural frequency is 3.75%. Increasing the thickness to 0.24 cm (3/32”) leads to increased error in the first modal frequency of 16.6%. It is established that for plate thickness at or below 0.16 cm (1/16”), the stand provides adequate clamping. Three different sandwich plates were tested. The first of these is a specimen, with aluminum face-plates of thickness 0.08 cm (1/32”) sandwiching a 0.00508 cm (2 mil) thick viscoelastic layer. The remaining two specimens are asymmetric, with face-plates of thickness 0.08 cm (1/32”) and 0.04 cm (1/64”). One of these plates has a core thickness of 0.00508 cm (2 mil), and the other has a core thickness of 0.0127 cm (5 mil). The viscoelastic material used is 3M Scotchdamp ISD-112. In our analysis, the GHM method is used to account for the complex modulus variation with frequency and temperature. The modulus and loss factors were obtained from the product information provided by 3M over the ranges of temperature and frequency of interest. In this case, the three mini-oscillators terms were used to fit the curve of storage modulus and loss factors. Table 4.4 shows the parameters used in the GHM method for a wide temperature range.

4.2.2.2

Analysis

For the “all sides clamped” boundary conditions, the plate transverse bending mode shapes in the w direction are obtained from beam bending modes. Analytical mode

93

Bolts Clamping Bar (3.81cm x 0.953cm) Sandwich plate Stand

Figure 4.2: Experimental setup for plate test shape of the Euler-Bernoulli beam in fixed-fixed end boundary conditions were given by Inman [27]. The plate bending mode shapes are the combinations of beam bending modes in x and y directions, so that w(x, y, t) =



W i (t)φm (x)φn (y)

(4.28)

i

where φm (x) = cosh(β m x) − cos(β m x) − λm (sinh(β m x) − sin(β m x)) λm =

cosh(β m l) − cos(β m l) sinh(β m l) − sin(β m l)

(4.29)

Here β m is determined using the characteristic equation for the fixed-fixed end boundary condition of a beam: cos(β m l) cosh(β m l) = 1

94

(4.30)

where l is the length of the beam. Similarly, we can obtain φn (y) by changing of length of beam and substituting x with y in Eq. 4.29. The approximate in-plane mode shapes assumed in face plates 1 and 3 for the case of a plate clamped on all sides are assumed to be mode shapes, as follows: u1 (x, y, t) =



Ui1 (t)Ψi (x, y)

i

v1 (x, y, t) =



V1i (t)Ψi (x, y)

i

u3 (x, y, t) =



U3i (t)Ψi (x, y)

i

v3 (x, y, t) =



V3i (t)Ψi (x, y)

(4.31)

i

The in-plane mode shapes for all the in-plane displacements were assumed as same: Ψi (x, y) = sin

nπy mπx sin L C

(4.32)

The in-plane mode shapes as represented by the rod mode shapes are different from those in simply-supported case because of the change in boundary conditions. For the transverse direction, 25 assumed modes were used in the model and 25 assumed modes were used in each of the in-plane directions. The effects of piezo-actuator were considered by providing the mass contributions of the piezo in our analysis. But, the stiffness contributions were neglected.

4.2.3

Results

The results of experiments conducted on the symmetric sandwich are tabulated against the frequencies predicted by the analysis in Table 4.6. Overall error in the 0 to 200 Hz frequency range is below 7%. But, if the bandwidth is increased, the errors are likely

95

to be higher. More bending and in-plane modes would need to be included to predict higher frequencies with more precision. The experimental validation of the asymmetric sandwich plates are presented in Table 4.7 and 4.8. Good correlation between measured and predicted modal frequencies is seen. A downward shift in the modal frequencies occurs when the thickness of the viscoelastic core increases. The trend is presented in both the experimental measurement of frequencies and the analytical predictions. The error increases for the higher modes. For the (4,1) mode of the symmetric sandwich plate, the error was 6.8%. For the asymmetric sandwich plates, the error in the (4,1) modal frequency was 10.5% for the 0.00508 cm (2 mil) case and 7.7% for the 0.0127 cm (5 mil) case. The viscous damping of some modes is also measured and compared with the predicted values for the symmetric plate in Table 4.6 and for the asymmetric plates in Table 4.7 and 4.8. Larger error is seen for the first modal loss factor implying the need for a more accurate damping model at lower frequencies. This error is also due to clamping conditions which causes additional surface friction over the clamped length. To examine the influence of the number of in-plane modes on the accuracy of modal frequency estimates in bending, the number of in-plane modes, ne , is varied while keeping the number of transverse vibration modes, nb , constant. These results are summarized in Table 4.9. The inclusion of the in-plane modes has a large impact on the analysis of sandwich plates. In-plane extension adds to the shearing of the viscoelastic core and therefore affects the overall stiffness of the sandwich. When ne = 1, the error is extremely large where the first modal frequency prediction has 112% error. By increasing ne , the error in prediction is reduced so that when ne = 6,

96

error in the first mode is down to 5.3%. The errors are still high for the higher modes. Once ne = 12, the frequencies of the first 6 modes agree well with the experimental results. When we decrease the number of bending modes from nb = 25 to nb = 18, the error for the first mode increases only from 3.95% to 4.21%. Thus, the shear strain in the viscoelastic core, is dominated by the in-plane modes and only mildly by the bending modes. Since the assumed bending and in-plane modes are orthonormal with themselves but not each other, accounting for the coupling between the in-plane and bending plate modes is crucial in sandwich structure analyses via the assumed modes method.

4.2.3.1

Influence of Operation Temperature

Based on the correlation between the experimental and analytical prediction in this section, we present a parametric study on the influence of operating temperature on the natural frequencies and the modal loss factors of the sandwich plate. The storage modulus and loss factor of viscoelastic materials are frequency and temperature dependent. The three plate specimens described above are subject to this study. The results of this study are presented in Figure 4.3 through Figure 4.5 where we plot frequency and loss factor vs. temperature for each plate. The curves have been plotted for the first five plate modes. The behavior of the curves is seen to be similar for different thicknesses of the viscoelastic layer (0.00508 cm or 0.0127 cm) (2 mil or 5 mil). The variation of temperature will cause the change of storage modulus and this has significant influence

97

on system stiffness, which affects both the modal frequencies and loss factors. The operating temperature of the plate needs to be considered when it is designed.

4.3

Summary

Analysis of sandwich plates with a dissipative core and isotropic face-plates was developed and validated. Transverse shear deformation of the face layers as well as the rotatory inertia are neglected. Flexural and membrane energies in the face-plates are accounted for, while the core is assumed to have shear stiffness alone. A first order shear deformation theory is used to describe the deformation in the layers. The core shear modulus is assumed to have a complex value which is dependent on the frequency. A traditional Galerkin assumed mode analysis of plate transverse bending was augmented with internal dissipation coordinates, using the GHM method, to account for the frequency dependent complex modulus of the viscoelastic core. We established the validity of our assumed modes analysis by comparison with an exact solution of a sandwich plate where the complex modulus of the viscoelastic core was assumed to be constant over the frequency range of interest. Validation of our analysis under simply-supported boundary conditions against this exact solution [28] shows an error of 0 and k22 > 0, then Xu = c1 sinh(k1 ζ) + c2 cosh(k1 ζ) + c3 sinh(k2 ζ) + c4 cosh(k2 ζ) Xv = d1 sinh(k1 ζ) + d2 cosh(k1 ζ) + d3 sinh(k2 ζ) + d4 cosh(k2 ζ)

2. if k12 > 0 and k22 < 0, then Xu = c1 sinh(k1 ζ) + c2 cosh(k1 ζ) + c3 sin(|k2 |ζ) + c4 cos(|k2 |ζ) Xv = d1 sinh(k1 ζ) + d2 cosh(k1 ζ) + d3 sin(|k2 |ζ) + d4 cos(|k2 |ζ) 3. if k12 < 0 and k22 < 0, then Xu = c1 sin(|k1 |ζ) + c2 cos(|k1 |ζ) + c3 sin(|k2 |ζ) + c4 cos(|k2 |ζ) Xv = d1 sin(|k1 |ζ) + d2 cos(|k1 |ζ) + d3 sin(|k2 |ζ) + d4 cos(|k2 |ζ) 4. if k12 = k22 < 0, then Xu = c1 sin(|k1 |ζ) + c2 cos(|k1 |ζ) + c3 ζ sin(|k2 |ζ) + c4 ζ cos(|k2 |ζ) Xv = d1 sin(|k1 |ζ) + d2 cos(|k1 |ζ) + d3 ζ sin(|k2 |ζ) + d4 ζ cos(|k2 |ζ) The sinh and cosh components correspond to near field (decay) waves and the sine and cosine components correspond to far field (propagation) waves as discussed by Doyle [17]. This mathematical representation of mode shapes matches the properties of wave propagation of in-plane plate vibration. For example, we consider a plate with a clamped edge at ζ = 0 and a free edge at ζ = 1. The expressions of mode shape functions Xu and Xv are assumed the same as the second case for k12 > 0 and k22 < 0. Substituting these functions into corresponding boundary conditions as shown in Eqs.

118

5.10 and 5.11 yields 

where

0 1 0 1     e1 0 −e2 0      p1 cosh(k1 ) p1 sinh(k1 ) p2 cos(k2 ) −p2 sin(k2 )   q1 sinh(k1 ) q1 cosh(k1 ) q2 sin(k2 ) q2 cos(k2 )





  c1       c   2  =0        c3      c4

(5.16)

k12 + a3 a2 k1 −k2 + a3 = − 2 a2 k2

e1 = − e2

p1 = a1 k1 + a22 e1 p2 = a1 k2 − a22 e2 q1 = b11 e1 k1 + b22 q2 = b11 e2 k2 + b22 Assuming non-trivial solutions of Eq. 5.16, the resulting four by four determinant is a function in terms of Ω only. First, we numerically determine an Ωx resulting in a zero determinant to obtain the modal frequency in ζ direction. Then the wave coefficients c1 , c2 , c3 and c4 can be solved for this particular frequency, Ωx . Finally, we can construct the mode shape functions Xu and Xv . The next step is to assume that Xu and Xv pair is prescribed a priori. Similarly we obtain: δu = Xu δYu δv = Xv δYv

(5.17)

Substituting Eqs. 5.3 and 5.17 into Eq. 5.2 and performing integration along ζ direction and following the same procedure as we did in the η directions. We show the

119

final two ODEs and associated boundary conditions. dYv d2 Yu + f2 + f 3 Yu = 0 2 dη dζ d2 Yv dYu g1 2 + g2 + g3 Yv = 0 dη dζ

f1

(5.18) (5.19)

And the boundary conditions at η = 0, 1 are needed. For a clamped edge: Yu = Yv = 0

(5.20)

For a free edge: dYv + g22 Yu = 0 dη dYu f11 + f22 Yv = 0 dη g1

(5.21) (5.22)

where f1 =


1

1 α2 (1 − ν)Xu2 dζ 2

f11 =


1

α2 Xu2 dζ


1

dXv 1 α(1 + ν) Xu dζ − να [Xv Xu ]1ζ=0 2 dζ


1

α

0

0

f2 =

0

f22 =

dXv Xu dζ dζ

0

f3 =


1  0

  1 d2 Xu dXu 2 2 Xu Xu + Ω Xu dζ − dζ 2 dζ ζ=0

120

(5.23)

g1 =


1

α2 Xv2 dζ


1

dXu 1 1 (1 + ν)α Xv dζ − (1 − ν)α [Xu Xv ]1ζ=0 2 dζ 2


1

α

0

g2 =

0

g22 =

dXu Xv dζ dζ

0

g3 =


1  0

 1  dXv d2 Xv 1 1 2 2 (1 − ν) Xv + Ω Xv dζ − (1 − ν) Xv 2 dζ 2 2 dζ ζ=0

(5.24)

We follow the same procedure to determine the modal frequency Ωy and the mode shapes Yu and Yv . We cannot find the exact solutions by applying the procedure only once for both ζ and η s because we wish to converge such that Ωix − Ωiy ≤ ǫ. Therefore, an iteration scheme is applied to achieve a convergent solution for both frequency and mode shapes. We summarize our iteration scheme as follows.

Step 1. In the η direction, prescribe the mode shape pair Yu0 and Yv0 a priori, k = 0.

Step 2a. Increment k, Yuk = Yuk−1 , and Yvk = Yvk−1

Step 2b. Obtain the ODEs in terms of Xuk and Xvk as shown in Eqs 5.8. Numerically solve for Ωkx that results in a zero determinant. The wave coefficients in Xuk and Xvk are determined under Ωkx .

Step 2c. Using mode shape function Xuk and Xvk as calculated in Step 2b, obtain the ODEs in terms of Yuk and Yvk as shown in Eqs. 5.18 Numerically solve for Ωky that

121

results in a zero determinant. The wave coefficients in Yuk and Yvk can be determined.

Step 3. Check convergence between Ωkx and Ωky . If Ωix − Ωiy ≤ ǫ, we stop the iteration. In our calculation, we set ǫ = 10−5 . Otherwise, go to step 2a, k = k + 1.

5.1.2

Validation and Results

Farag and Pan [21] considered three rectangular plates, as shown in Figure 5.2, CCCC, CCCF and CFCF cases. Plate dimensions are 1.0 m in length and 1.2 m in width and 2.5 mm thick. The Young’s modulus is E = 70 × 109 N/M 2 , density is 2700kg/m3 , and the Poisson’s ratio is ν = 0.33. In order to validate this method, we will calculate natural frequencies and mode shapes for the in-plane plate vibration problems investigated by them. In order to start our approach, we need to discuss how to chose the initial assumed mode shape pair in either ζ or η direction. General speaking, those mode shapes must satisfy two conditions. First, they should be admissible functions, which satisfy the geometric boundary conditions. Secondly, the resulting coupling coefficients in Eq. 5.6 or in Eq. 5.23, that is, a2 or f2 , cannot be zero. This ensures that we solve the coupled mode shapes as discussed in the previous section. The one-dimensional rod vibration mode shapes are used to initialize the iteration calculation and are tabulated for different boundary conditions in Table 5.1.3. These rod mode shape functions are admissible functions in the x and y directions. Because of the orthogonality of trigonometric functions as shown in the table, the summation of modal number for the

122

assumed pair of Yum and Yvn has to be an odd number in order to satisfy the second criterion for the initial mode pair. The condition mod(m+n)=1 must hold to achieve non-zero value of a2 or f2 and our numerical calculation results validate this remark. We tabulated the first six natural frequencies of in-plane rectangular plate vibration where all three boundary condition cases were considered. Table 5.2 shows the natural frequencies of a clamped rectangular plate (CCCC) case. Our results are compared to the solutions of NASTRAN and to those of Farag and Pan [21]. Compared to NASTRAN results, the maximum error is 1.6% in our analysis and 4.6% for Farag and Pan’s. Thus, improved accuracy has been achieved in our analysis because the plate mode shapes are more accurate. Table 5.3 shows the frequency results of the CCCF case. The error increases for both analyses compared to NATRAN because a free edge is introduced. The maximum error is 3.9% in our analysis and 8.4% for Farag and Pan’s. Table 5.4 shows the frequency results of the CFCF case where our results are an improvement over Farag and Pan’s. The maximum error in Farag’s results reaches 12 % compared to 4.5% in our analysis. We demonstrate mode shape functions of the first six modes shapes in Figure 5.3 to 5.5 for all three cases. These are plotted in vector form where the origin of arrow denotes the location and the length of the arrow denotes the magnitude of the resultant displacements giving us a visualization of the mode shapes for in-plane plate motion. As shown in Figure 5.3 for the CCCC case, we can observe some node lines in which displacement is dominant only in one direction. The displacements are symmetric with respect to those lines for the first, second, fifth and sixth mode. For the third and fourth modes, the shear (rotation) mode shape is easily identified. The third mode corresponds to rotation with respect

123

to the center of a plate. For the fourth mode, the displacements behave similarly to the case in which the extension and compression occurs along two diagonals of a plate. For the CCCF case, as shown in Figure 5.4, the mode shape displacements show the different results compared to the CCCC case. Due to an introduction of a free edge, the displacements are small close to clamped edges and become larger when reaching the free edge. The mode shapes only show symmetries along the y direction. For the CFCF case, as shown in Figure 5.5 similar results are obtained compared to the CCCC case. We can identify the node lines easily and the axes of symmetry.

5.1.3

Summary

Based on the Kantorovich method, we computed the natural frequencies and natural modes of rectangular plates. The analytical results were validated using both NASTRAN and results from literature [21]. Improved accuracy for the natural frequency calculations for three cases was achieved when compared to available results from literature relative to the NASTRAN analysis. Mode shapes were expressed as a linear combination of wave propagation where the wave coefficients were computed using a numerical iteration scheme. The mode shapes were given in analytical form in which the wave coefficients were determined through a numerical iteration scheme. 1. As shown in Table 5.2 for CCCC case, the maximum error of our analysis was 1.6% and 4.6% for Farag and Pan’s analysis. Both analyses predict natural frequency well. 2. As shown in Table 5.3 for CCCF case, the maximum error of our analysis was

124

Table 5.1: Admissible rod mode shape functions Boundary Conditions

Mode Shape Function

clamped-clamped

) Wm = sin( mπx l

clamped-free

Wm = sin( (2m−1)πx ) 2l

free-free

Wm = cos( mπx ) l

3.9% and 8.4% for Farag and Pan’s analysis. The errors in the CCCF case increase for both analyses. The introduction of a free edge increases the displacement coupling because a force boundary condition exists. 3. As shown in Table 5.4 for CFCF case, the maximum error continues to increase, 4.5% in our analysis and 12% in Farag and Pan’s. As more free edges are added, the coupling effects between mode shapes increase. Overall more accurate approximation of frequency calculations were achieved in our analysis relative to the NASTRAN analysis. The plots of mode shapes provide us a visualization of displacement field for in-plane plate vibration.

125

Table 5.2: Natural frequencies of in-plane vibration of a rectangular plate with CCCC boundary conditions

Mode

Mode

NASTRAN

for u

for v

Freq.

Freq.

Error

Freq.

Error

m×n m×n

[Hz]

[Hz]

[%]

[Hz]

[%]

1

2×2

1×1

2658

2667

0.3

2671

0.5

2

1×1

2×2

2898

2909

0.4

2914

0.6

3

1×2

2×1

3260

3280

0.6

3349

2.7

4

1×2

2×1

4024

4089

1.6

4198

4.3

5

1×3

2×2

4268

4327

1.4

4404

3.2

6

2×3

1×2

4404

4437

0.7

4607

4.6

Mode No.

126

Present

Farag and Pan

Table 5.3: Natural frequencies of in-plane vibration of a rectangular plate with CCCF boundary conditions

Mode

Mode

NASTRAN

for u

for v

Freq.

Freq.

Error

Freq.

Error

m×n m×n

[Hz]

[Hz]

[%]

[Hz]

[%]

1

2×2

1×1

1803

1811

0.4

1892

4.9

2

1×1

2×2

2656

2674

0.7

2727

2.7

3

1×2

2×1

2794

2845

1.8

3026

8.4

4

1×2

2×1

3392

3524

3.9

3596

6.0

5

2×3

1×2

3479

3504

0.7

3624

4.2

6

1×3

2×2

3704

3757

1.4

3868

4.4

Mode No.

127

Present

Farag and Pan

Table 5.4: Natural frequencies of in-plane vibration of a rectangular plate with CFCF boundary conditions; Modal number 0 denotes the ”rigid” mode.

Mode

Mode

NASTRAN

for u

for v

Freq.

Freq.

Error

Freq.

Error

m×n m×n

[Hz]

[Hz]

[%]

[Hz]

[%]

1

2×3

1×0

1449

1455

0.4

1531

7.0

2

2×2

1×1

2511

2520

0.4

2682

6.0

3

1×0

2×1

2567

2639

2.8

2697

5.0

4

1×1

2×0

2637

2662

0.95

2994

12.0

5

1×1

2×0

3037

3187

4.5

3122

3.0

6

1×2

2×1

3061

3146

2.8

3390

10.0

Mode No.

128

Present

Farag and Pan

y

CCCC

clamped

clamped

clamped

x clamped y

CCCF

clamped

clamped

free

x clamped

CFCF

clamped

clamped

y free

x free

Figure 5.2: Three configurations of rectangular plate under in-plane vibration

129

1.2

1

1

0.8

0.8

y,v

y,v

1.2

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

x,u

1.2

1.2

1

1

0.8

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

x,u

1

1

0.8

0.8

y,v

y,v

1.2

0.6

1

0.6

0.4

0.4

0.2

0.2

0.4

0.8

d) ω = 4089Hz u(1 × 2); v(2 × 1)

1.2

0.2

0.6 x,u

c) ω = 3280Hz u(1 × 2); v(2 × 1)

0 0

1

0.6

0.4

0.2

0.8

b) ω = 2909Hz u(1 × 1); v(2 × 2)

y,v

y,v

a) ω = 2667Hz u(2 × 2); v(1 × 1)

0 0

0.6 x,u

0.6

0.8

1

0 0

0.2

x,u

e) ω = 4327Hz u(1 × 3); v(2 × 2)

0.4

0.6

0.8

1

x,u

f) ω = 4437Hz u(2 × 3); v(1 × 2)

Figure 5.3: Mode shapes of in-plane vibration of a rectangular plate with CCCC boundary conditions

130

1.2

1

1

0.8

0.8

y,v

y,v

1.2

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

x,u

1.2

1.2

1

1

0.8

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

x,u

1

1

0.8

0.8

y,v

y,v

1.2

0.6

1

0.6

0.4

0.4

0.2

0.2

0.4

0.8

d) ω = 3524Hz u(1 × 2); v(2 × 1)

1.2

0.2

0.6 x,u

c) ω = 2845Hz u(1 × 2); v(2 × 1)

0 0

1

0.6

0.4

0.2

0.8

b) ω = 2674Hz u(1 × 1); v(2 × 2)

y,v

y,v

a) ω = 1811Hz u(2 × 2); v(1 × 1)

0 0

0.6 x,u

0.6

0.8

1

0 0

0.2

x,u

e) ω = 3504Hz u(2 × 3); v(1 × 2)

0.4

0.6

0.8

1

x,u

f) ω = 3757Hz u(1 × 3); v(2 × 2)

Figure 5.4: Mode shapes of in-plane vibration of a rectangular plate with CCCF boundary conditions

131

1.2

1

1

0.8

0.8

y,v

y,v

1.2

0.6

0.6

0.4

0.4

0.2

0.2

0 0

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

x,u

1.2

1.2

1

1

0.8

0.8

0.6

0.4

0.2

0.2

0.4

0.6

0.8

0 0

1

0.2

0.4

x,u

1

1

0.8

0.8

y,v

y,v

1.2

0.6

1

0.6

0.4

0.4

0.2

0.2

0.4

0.8

d) ω = 2662Hz u(1 × 1); v(2 × 0)

1.2

0.2

0.6 x,u

c) ω = 2639Hz u(1 × 0); v(2 × 1)

0 0

1

0.6

0.4

0.2

0.8

b) ω = 2520Hz u(2 × 2); v(1 × 1)

y,v

y,v

a) ω = 1455Hz u(2 × 3); v(1 × 0)

0 0

0.6 x,u

0.6

0.8

1

0 0

0.2

x,u

e) ω = 3187Hz u(1 × 1); v(2 × 0)

0.4

0.6

0.8

1

x,u

f) ω = 3146Hz u(1 × 2); v(2 × 1)

Figure 5.5: Mode shapes of in-plane vibration of a rectangular plate with CFCF boundary conditions

132

z

C y

L

w Mxy Qyz Myy Mxx

Qxz

Mxy

x

Figure 5.6: Schematic of rectangular plate bending vibration

5.2

Plate Bending Vibration

In this section, we will discuss the mode shapes of isotropic plate bending vibration. Typically beam bending mode shapes are used in both x and y directions to approximate the mode shapes of plate bending through the Rayleigh-Ritz method [35] [9]. These approximation provided upper bounds of natural frequency calculations in which many numbers of mode shapes were included. Bhat et. al. [8] solved bending mode shapes of rectangular plate having simply-supported and clamped boundary conditions. We will use the same method to solve the mode shapes of rectangular plate with clamped and free boundary conditions. Details of this method were given in the paper of Bhat et. al. [8]. A thin plate with Kirchhoff hypothesis is considered and the corresponding non-dimensional variational expression of total energy of the

133

plate, as shown in Figure 5.6, is: δU

=


1
1  0

+

0


1  0

−


1 0

 4 4 ∂4W 2 ∂ W 4∂ W 2 + 2α +α + Ω W δW dζdη ∂ζ 4 ∂ζ 2 ∂η 2 ∂η 4

 
1 
1 ∂W 1 ∂W 1 Mζ δ( ) dη + Mη δ( ) dζ − [Vζ δW ]10 dη ∂ζ 0 ∂η 0 0

[Vη δW ]10 dζ −



0

2(1 − µ)α2

∂2W δW ∂ζ∂η

1 1

=0

(5.25)

0 0

Here we define some non-dimensional parameters. The non-dimensional length in x and y directions are: ζ=

x L

η=

y C

The aspect ratio of plate and non-dimensional frequency are: α=

L C

Ω2 = ω 2



mVL4 EI



where m is mass per unit area of a plate and EI is the plate bending flexural stiffness. The non-dimensional resultant shear forces and moments are: Vζ Vη Mζ Mη

∂3w ∂3w 2 + α (2 − ν) ∂ζ 3 ∂ζ∂η 2 3 ∂ w ∂3w = α4 3 + α2 (2 − ν) 2 ∂η ∂ζ ∂η 2 2 ∂ w ∂ w = + να2 2 2 ∂ζ ∂η 2w ∂ ∂2w = α4 2 + να2 2 ∂η ∂ζ =

The separable solution of bending mode shape function, Wmn , is assumed as: Wmn (ζ, η) = Xm (ζ)Yn (η)

(5.28)

δWmn (ζ, η) = Yn (η)δXm (ζ)

(5.29)

If we assume Yn a priori

134

Substitute above two equations into Eq. 5.25 and integrate along η direction, this yields: d2 Xm d4 Xm + 2β + γx Xm = 0 x dζ 4 dζ 2

(5.30)

where βx is a constant and γx is a function of unknown parameter Ω. The expression of these two parameters are:

where

βx =

1

γx =

1

0

0

α2 Yn′′ Yn dη + α2 (ν − 1) [Yn′ Yn ]10 1 2 0 Yn dη

! α4 Yn′′′′ Yn dη + α4 [Yn′′ Y )n′ ]10 − [Yn′′′ Yn ]10 − Ω2 1 2 dη Y 0 n ()′ =

d () dη

The solution of the above Eq. 5.30 is: Xm = c1 sin(px ζ) + c2 cos(px ζ) + c3 sinh(qx ζ) + c4 cosh(qx ζ)

(5.32)

where px qx

" "  " " 2 = "" −βx − βx − 4γx "" " "  " " 2 " = " −βx + βx − 4γx ""

px and qx are wave numbers, which are functions of Ω. The coefficients of c1 , c2 , c3 , c4 were determined using boundary condition at two edges ζ = 0, and ζ = 1. The possible boundary condition on these two edges are; (a) clamped edge Xm = 0,

dXm =0 dζ

Xm = 0,

d2 Xm =0 dζ 2

(b) simply-supported edge

135

(c) free edge d2 Xm + e1 Xm = 0 dζ 2 d3 Xm dXm = 0 + e2 3 dζ dζ where e1 = e2 =

 1 ′′ 2 0 Yn Yn dη να  1 2 0 Yn dη  1 ′′ Y Yn dη α2 (2 − ν) 0 1 n 2 0 Yn dη

[Y ′ Yn ]1 − 2α2 (1 − ν)  1n 0 2 0 Yn dη

The next step is to determine wave coefficients, c1 , c2 , c3 , c4 . For example, the case in which a plate is clamped at ζ = 0 and free at ζ = 1, then  

where

0 1 0 1     px 0 qx 0      f1 sin(px ) f1 cos(px ) f2 sinh(qx ) f2 cosh(qx )   f3 cos(px ) f4 cos(px ) f5 cosh(qx ) f5 sinh(qx )



  c1       c   2   =0       c3      c4

(5.35)

f1 = −p2x + e1 f2 = qx2 + e1 f3 = −p3x + e2 px f4 = p3x − e2 px f5 = qx3 + e2 qx The non-trivial solutions of those coefficients will lead to a frequency equation in terms of Ω only. Eq. 5.35 is solved to obtain an Ωxm and the coefficients in mode shape

136

function Xm are determined. Alternatively, we can obtained functions Xm as a priori, then δWmn (ζ, η) = Xm (ζ)δYn (η)

(5.36)

We substitute the above equation into Eq. 5.25 and integrate along ζ direction which yields an ODEs in terms of Yn . d2 Yn d4 Yn + 2β + γy Yn = 0 y dη 4 dη 2

(5.37)

where βy is constant and γy is a function of unknown parameter Ω. The expression of these two parameters are:

where

βy =

1

γy =

1

0

0

′′ X dζ + α2 (ν − 1) [X ′ X ]1 α2 Xm m m m 0 1 4 2 α 0 Xm dζ

! ′′′′ X dζ + [X ′′ X ′ ]1 − [X ′′′ X ]1 Xm m m m 0 m m 0 Ω2 − 4 1 2 dζ α α4 0 Xm ()′ =

d () dζ

The solution of the above equations is: Yn = d1 sin(py η) + d2 cos(py η) + d3 sinh(qy η) + d4 cosh(qy η)

(5.39)

Similarly, the wave numbers were given by: py qy

" " " "  " " = " −βy − βy2 − 4γy " " " " " " "  " " = " −βy + βy2 − 4γy " " "

The possible boundary conditions along two edges η = 1, 1 are; (a). clamped edge Yn = 0,

dYn =0 dη

137

(b). simply-supported edge Yn = 0,

d2 Yn =0 dη 2

(c). free edge d2 Yn + g1 Yn = 0 dη 2 d3 Yn dYn + g2 = 0 dη 3 dη where ′′ 0 Xm Xm dζ 1 2 dζ α2 0 Xm  1 ′′ X Xm dζ (2 − ν) 0  1 m 2 0 Yn dη

g1 = ν g2 =

1

′ X ]1 [Xm m 0 − 2(1 − ν)  1 2 2 dζ α 0 Xm

By applying boundary conditions at two edges η = 0, 1, we can solve a frequency equation to obtain an Ωyn and determine the coefficients in mode shape function Yn . We will repeat the whole procedure until convergence is achieved for frequency in both x and y directions, ie., Ωxm − Ωyn ≤ ǫ (ǫ = 1.0e−5 in our calculation). Then the mode frequency Ωmn and mode shapes Wmn are determined. Validation of this method for plate bending vibration is presented in Chapter 6.

138

y

Plate specimen With CFCF BC.

10 in.

x 12 in.

Figure 5.7: A uniform rectangular plate with CFCF boundary conditions

5.3

Results for Plate Bending and In-plane Mode Shape Functions

In this section, we will calculate the plate mode shape functions for bending and in-plane vibration for a uniform rectangular plate. The boundary conditions were clamped on two parallel edges along x direction and free on the two edges along y direction, which is denoted by CFCF is shown in Figure 5.7. The aspect ratio of length in x and y direction is 1.2. And the Poisson ratio was assumed ν = 0.3. Table 5.5 and 5.6 showed the plate bending mode shape functions. Plate in-plane mode shape functions were presented in Table 5.7 to 5.10 for both displacements u and v. These mode shapes will be used in next chapter for the analyses of sandwich plates.

139

Table 5.5: The parameters in mode shape functions of a rectangular plate bending vibration under CFCF boundary condition I: where Wmn (x, y) = Xwm Ywn

mn

        Xwm = sin p1 xl + c1 cos p1 xl + c2 sinh p2 xl + c3 cosh p2 xl c1

c2

c3

p1

p2

11

-1.0178

-1

1.0178

4.73

4.73

12

-0.66715 -0.66513 0.66715 4.3183

6.4924

13

-0.34212 -0.34211 0.34212 3.8009

11.11

21

-0.99922

0.99922 7.8532

7.8532

22

-0.85305 -0.85325 0.85305 7.6957

9.0192

23

-0.58028 -0.58028 0.58028 7.3348

12.64

14

-0.21739 -0.21739 0.21739 3.5697

16.421

31

10.996

10.996

32

-0.92388 -0.92387 0.92388 10.916

11.816

24

-0.40644 -0.40644 0.40644 7.0553

17.359

33

-0.72389 -0.72389 0.72389 10.678

14.751

15

-0.15898 -0.15898 0.15898 3.4569

21.744

41

-1

-1

-1

-1

-1

1

1

14.137

14.137

42

-0.95631 -0.95631 0.95631 14.093

14.736

34

-0.55348 -0.55348 0.55348 10.436

18.855

25

-0.30647 -0.30647 0.30647 6.8779

22.443

140

Table 5.6: The parameters in mode shape functions of a rectangular plate bending vibration under CFCF boundary condition II: where Wmn (x, y) = Xwm Ywn

mn

        Ywm = d1 sin q1 yc + d2 cos q1 yc + d3 sinh q2 yc + d4 cosh q2 yc d1

d2

d3

d4

q1

q2

11

0

5.6667

-0.97247

1

5.9605e-8

4.1336

12

1

-2.8336

1.2201

-1.1996

2.4631

4.7684

13

1

-1.5252

1.0957

-1.099

5.1225

6.4878

21

0

5.6667

-0.9995

1

5.9006e-7

7.9974

22

1

-4.2674

1.1673

-1.1668

2.6812

8.4014

23

1

-2.507

1.2002

-1.2004

5.5241

9.6129

14

1

-1.2092

1.043

-1.0427

8.0428

8.9421

31

0

5.6667

-0.99999

1

5.4355e-6

11.72

32

1

-4.4398

0.99852

-0.99851

2.6985

12.016

24

1

-1.7252

1.1244

-1.1244

8.3741

11.427

33

1

-3.4344

1.2296

-1.2296

5.7165

12.985

15

1

-1.1083

1.0231

-1.0231

11.098

11.751

41

0

5.6667

-1

1

2.5194e-5

15.437

42

1

-4.0701

0.83143

-0.83143

2.6597

15.66

34

1

-2.3739

1.1907

-1.1907

8.6274

14.446

25

1

-1.4036

1.0773

-1.0773

11.328

13.701

141

Table 5.7: The parameters in mode shape functions of a rectangular plate inplane vibration under CFCF boundary condition I: where Umn (x, y) = Xum Yun

Xum I II m

Index

2

        Xum = c1 sinh p xl + c2 cosh p xl + c3 sin q xl + c4 cos q xl         Xum = c1 sin p xl + c2 cos p xl + c3 sin q xl + c4 cos q xl c1

c2

c3

c4

p

q

I

1.0169

-1

0.20688

1

4.7806

3.5496

1

I

0.93906

-1

-14.588

1

3.4601

3.2785

1

II

1

0.75982

0.11476

-0.75982

1.8421

5.9834

2

I

2.7646

-1

4.9485

1

1

II

1

-0.0656

0.010374

0.0656

3.2726

1

II

1

2.4638

0.28628

-2.4638

0.77111 6.0518

4

I

1.0111

-1

0.10615

1

5.1964

9.6363

1

II

1

-0.81836

0.44581

0.81836

4.5133

7.2808

2

II

1

-0.6023

0.11295

0.6023

1.0842

9.7955

1

II

1

-0.072948

0.15154

0.072948

3.2872

8.5275

2

I

1.0095

-1

-3.655

1

5.3535

6.8173

2

I

1.001

-1

-34.625

1

7.6214

6.3409

2

II

1

0.20008

0.13964

-0.20008

5.8882

8.2061

1

I

0.73171

-1

-0.5222

1

1.8648

5.3207

5

I

0.99939

-1

0.11391

1

8.0949

12.793

1

II

1

0.59293

0.033994

-0.59293

2.0712

12.452

142

0.75771 5.8844 6.5969

Table 5.8: The parameters in mode shape functions of a rectangular plate inplane vibration under CFCF boundary condition II: where Umn (x, y) = Xum Yun

Yun I II III IV

        Yun = c1 sinh p yc + c2 cosh p yc + c3 sin q yc + c4 cos q yc         Yun = c1 sin p yc + c2 cos p yc + c3 sin q yc + c4 cos q yc         Yun = c1 sinh p yc + c2 cosh p yc + c3 sinh q yc + c4 cosh q yc         Yun = c1 sin p yc + c2 cos p yc + c3 yc sin q yc + c4 yc cos q yc

n

Index

c1

4

III

1

1

IV

2

c2

c3

c4

p

q

-0.99955 -0.15398

0.054059

8.4081

0.73334

1

0.84322

-0.18094

0.53701

1.1732

1.1732

II

1

-0.64692

0.70195

-0.87644

1.1484

1.791

3

I

1

-1.0035

0.95143

0.072566

6.3638

2.9893

2

II

1

-0.63965

-1.8536

-2.645

1.1381

4.3641

3

II

1

1.0508

-1.5325

2.5614

1.5212

5.2048

4

III

1

-0.99599 -0.58362

0.52265

6.209

2.8984

1

I

1

-1.6341

-1.3

-0.0024262

1.4241

3.1379

3

I

1

-1.0151

0.54466

0.40004

4.8914

1.8746

3

II

1

-0.21141

-1.266

-3.6229

3.5583

6.9556

2

I

1

-0.65239

-1.2709

-0.17256

1.5589

6.0133

1

I

1

-8.8511

-0.14415

-3.4811

0.22693

6.366

1

II

1

1.8414

-0.79962

1.0853

0.99499

5.0132

4

II

1

1.372

-1.2732

8.8358

4.4012

9.1386

4

I

1

-0.99977 -0.51028

0.48291

9.0571

3.5916

4

I

1

-0.60493

-2.9613

1.4018

8.0594

2.4078

143

Table 5.9: The parameters in mode shape functions of a rectangular plate inplane vibration under CFCF boundary condition III: where Vmn (x, y) = Xvm Yvn

Xvm

m Index

        Xvm = c1 sinh p xl + c2 cosh p xl + c3 sin q xl + c4 cos q xl         Xvm = c1 sin p xl + c2 cos p xl + c3 sin q xl + c4 cos q xl c1

c2

c3

c4

p

q

1

I

-0.92197

0.93757

4.532

-0.93757

4.7806

3.5496

2

I

-8.5959

8.072

-0.55334

-8.072

3.4601

3.2785

2

II

-0.20923

0.27537

-1.8232

-0.27537

1.8421

5.9834

1

I

8.6376

-23.88

-4.8256

23.88

2

II

0.51895

-0.082069

3.2726

2

II

-1.6694

0.67758

-5.8316

-0.67758

0.77111 6.0518

3

I

-0.058056

0.058702

0.55301

-0.058702

5.1964

9.6363

2

II

-1.1906

-1.4548

-2.6706

1.4548

4.5133

7.2808

3

II

0.19865

0.32982

1.7587

-0.32982

1.0842

9.7955

2

II

0.032953

0.45173

0.21745

-0.45173

3.2872

8.5275

3

I

-1.7969

1.814

-0.4963

-1.814

5.3535

6.8173

3

I

-17.039

17.056

-0.49258

-17.056

7.6214

6.3409

3

II

0.19377

-0.96847

1.3876

0.96847

5.8882

8.2061

2

I

0.81315

-0.59498

1.1394

0.59498

1.8648

5.3207

4

I

-0.014857

0.014847

0.13034

-0.014847

8.0949

12.793

4

II

-0.099851

0.1684

-2.9373

-0.1684

2.0712

12.452

I II

0.0053838 0.082069

144

0.75771 5.8844 6.5969

Table 5.10: The parameters in mode shape functions of a rectangular plate inplane vibration under CFCF boundary condition IV: where Vmn (x, y) = Xvm Yvn

Yvn I II III IV n

Index

1

        Yvn = c1 sinh p yc + c2 cosh p yc + c3 sin q yc + c4 cos q yc         Yvn = c1 sin p yc + c2 cos p yc + c3 sin q yc + c4 cos q yc         Yvn = c1 sinh p yc + c2 cosh p yc + c3 sinh q yc + c4 cosh q yc         Yvn = c1 sin p yc + c2 cos p yc + c3 yc sin q yc + c4 yc cos q yc c1

c2

c3

c4

p

q

III

-0.022032

0.022041

0.14308

-0.40755

8.4081

0.73334

2

IV

-2.1465

2.1107

-0.946

-0.31875

1.1732

1.1732

1

II

-0.3648

-0.56391

-0.5197

-0.41623

1.1484

1.791

2

I

0.027487

-0.027392

-0.02889

0.37878

6.3638

2.9893

1

II

-2.7986

-4.3752

1.5098

-1.0581

1.1381

4.3641

2

II

-1.3734

1.3069

0.89616

0.53619

1.5212

5.2048

1

III

-1.3063

1.3116

2.2384

-2.4995

6.209

2.8984

2

I

-2.1714

1.4241

3.1379

2

I

-0.26189

0.25799

0.48083

-0.65466

4.8914

1.8746

2

II

-1.0299

-4.8719

0.81352

-0.28429

3.5583

6.9556

3

I

0.44711

-0.68534

0.35179

-2.591

1.5589

6.0133

4

I

0.96586

-0.10912

6.7003

-0.27746 0.22693

6.366

2

II

0.6751

-0.36662

-0.84427

-0.62202 0.99499

5.0132

3

II

-5.0822

3.7042

3.9917

0.57518

4.4012

9.1386

1

I

-5.2303

5.2316

10.25

-10.831

9.0571

3.5916

1

I

1.2664

-2.0935

-0.52598

-0.42767

1.4018

8.0594

0.0052686 -0.0032241 0.0040526

145

Chapter 6

Analyses of Sandwich Plate: Part II

In the previous chapter, we used the Kantorovich method to solve the plate bending and in-plane vibration problem for rectangular plates, where the mode shape functions were solved in the closed form. For the plate in-plane vibration, the NASTRAN results and results from literature was used to validate our analysis. Our results were comparable to the NASTRAN results for the frequency predictions. Our goal here is to apply those higher order plate mode shapes functions in the analysis of sandwich plates with a viscoelastic core. We have conducted experiments to test an aluminum plate with and without partial PCLD treatment. For the aluminum plate, we can validate the results of plate bending mode shape functions solved using the Kantorovich method. For the plate with partial PCLD treatment, we try to improve our sandwich plate analysis using plate modes in order to include fewer number of modes. A thorough validation will be done for the predictions of natural frequency, loss factor, mode shapes, and frequency response functions using testing data. The configuration of our two plates was clamped on the two edges in x direction, and free

146

on two edges in y direction, denoted by CFCF.

6.1

Experimental Set-up

Figure 6.1 shows the experimental set-up. A shaker was used as the excitation source, which hung about 15 inches away above the plate. The reason for this is to minimize the influence of the shaker. If we fix the shaker, there is additional stiffness contribution from the shaker, which will change the properties of the whole system, and this effect is difficult to include in our analysis. Measured natural frequencies of lower modes using an impact hammer were similar to the results under the shaker excitation for an aluminum plate. Therefore, we do not need to include the effects of shaker in our analysis. The force output from shaker was transmitted through a load cell and a rigid rod to the plate, as shown in Figure 6.1. The rigid rod was bonded to the surface of plate using M-bond and provided a good adhesion between the rod and plate. The size of the rod is about 1.5 inch long and 5/16 inch diameter. The load cell provided the magnitude of force input to the plate. When we assemble the whole system, we have to make sure the rigid rod is perpendicular to the surface of the plate in order to introduce a vertical point force only. We normally let the rod just touch the surface of the plate by adjusting the length of elastic strings and the glue will fill the gap between the tip of the rod and plate providing a point transverse force input to the plate. As shown in Figure 6.1, our base is an optical table with a vibration isolation

147

system. This isolation workstation is made by Newport Corporation. The optical table is RS-3000 with honey cone and integrated tuned damping. The top surface is 400 series ferromagnetic stainless steel with 1 inch by 1 inch screw pattern, the diameter is 5/16 inch. An air compressor is served as the air source to the isolation legs of the workstation system. The isolation system floats the table and very low frequency disturbances from floor were totally isolated. The details of this was given by Ryaboy al el [55]. A plate was clamped by fixtures on two parallel edges and free on the other two edges, as shown in Figure 6.1. The fixtures were designed to provide clamped boundary conditions and were made of top and bottom parts. The size of top and bottom parts were the same, 15 in. long, 3 in. wide, and 1 inch thick. The two bottom pieces were bolted to the optical table with a distance 12 in. apart. A 13 by 10 in. aluminum plate specimen with a thickness of 0.05975 in, was placed on atop of the two bottom pieces. The clamping width was a half inch at each clamped edge. Two strips with same thickness as the plate were placed on the each of the back edges of bottom pieces. Then the top pieces of the fixture were bolted to the bottom pieces. A torque of 200 in-lbs was applied to each of the bolts to provide uniform clamping. Figure 6.2 displays the details of the clamping fixture. Two specimens with CFCF boundary conditions were tested. One was an aluminum plate and the other was an aluminum plate with partial PCLD treatment. The size of aluminum plate was 12 in. long, 10 in. wide, and 0.05975 in. thick. The plate with PCLD was 12 in. long, 10 in wide, and the base plate thickness was 0.05975 in. The constraining plate layer was 4 in. long, 10 in. wide, and 0.015 in. thick. The

148

viscoelastic core is 3M ISD112 with 2 mil thickness. The PCLD treatment was located on the center of base plate, which was from x1 = 4 in. to x2 = 8 in., as shown in Figure 6.3. The aluminum material was 6061T6 with the Young’s Modulus E = 68GP a and Poisson ratio ν = 0.3. As shown in Figure 6.1, a non-contact Schaevitz DistanceStar laser sensor was used to measure the displacement of the plate. We obtained a frequency response function at fifteen positions on the plate. The coordinates of these measurements were listed in Table 6.1 and 6.1 for the aluminum plate and the plate with PCLD, respectively. We chose the excitation location carefully to avoid exciation at the nodal position in the plates. We draw the lines which equally divided the length in both x and y directions. Then, we can find the location which can excite up to fourth mode in each direction. This simple scheme was validated by our experiment. In order to reach up to mode (1,4), the excitation location was located at (x, y) = 10 85 , 2 78 in. for the aluminum plate and (x, y) = 10 85 , 2 in. for the aluminum plate with PCLD. A sine sweep signal was applied to the shaker with the load cell feedback to maintain the constant force magnitude for the whole frequency spectra. A similar setup, as shown in Figure 3.7, was used for plate test except the load cell feedback control scheme. We set up the control voltage in input of the load cell and the output voltage to shaker were adjusted based on the feedback control algorithm integrated in the Siglab signal acquisition system.

149

Laser Sensor Shaker Load cell Specimen Optical Table

Figure 6.1: Schematic of plate testing set-up

Top Piece

Plate Specimen

Shim

1 in.

Bottom Piece 0.5 in.

0.5 in. 3 in.

Figure 6.2: Diagram of clamping fixture

150

y 12 in.

10 in.

x X1=4 in. X2=8 in.

Plate with PCLD Treatment

Figure 6.3: A plate with PCLD treatment under CFCF boundary conditions

1

6

11

2

7

12

3

8

13

4

9

14

5

10

15

Figure 6.4: Schematic of sensor array for plate testing

151

Table 6.1: Coordinates of the 15 measured locations for an aluminum plate under CFCF boundary conditions; x and y are in inches 1

2

3

4

5

x

3

3

2 31 32

15 2 16

2 15 16

y

9 87

7.5

5

17 2 32

1 8

6

7

8

9

10

x

6

6

6

6

6

y

9 87

15 7 32

5

7 2 16

1 8

11

12

13

14

15

x 7.5

7.5

17 7 32

7.5

7.5

9 87

17 7 32

5

2.5

1 8

y

Table 6.2: Coordinates of the 15 measured locations for a plate with PCLD treatment under CFCF boundary conditions; x and y are in inches 1

2

3

4

5

x

3

3

3

2 31 32

15 2 16

y

15 9 16

7.5

5

2.5

1 16

6

7

8

9

10

x

6

6

6

5 15 16

5 78

y

9 87

15 7 32

5

2.5

3 32

11

12

13

14

15

9 x 7 16

17 7 32

7 17 32

7.5

7.5

29 9 32

7.5

5

2.5

1 32

y

152

6.2

Results

In this section we will show the results of frequency, mode shape functions and frequency response function for both the aluminum plate and the aluminum plate with PCLD treatment. As shown in the Section 5.3, the first 16 plate bending and in-plane mode shapes were solved under the CFCF boundary conditions and were presented in Table 5.5 to Table 5.10. These plate modes will be used in our analysis. The dimensions of the two specimen were discussed in Section 6.1. The material constant for Young’s Modulus was 68GP a and the Poisson ratio was assumed as ν = 0.3. The viscoelastic core was working under room temperature, at 20o C.

6.2.1

Aluminum Plate

Based on the experimental frequency response functions at 15 locations on the plate specimen, we can extract the first seven modal frequencies, modal damping, and mode shape functions using the Star software[72]. These results were used to validate our analysis of an aluminum plate. In our analysis, the assumed modes method was employed to solve the aluminum plate vibration problem using either beam bending mode shapes or the plate mode shapes as shown in Table 5.5 and 5.6. The transverse displacement w(x, y) was assumed as an expansion of beam mode shape function in both x and y direction. w(x, y) =



n Wi (t)φm w (x)φw (y)

(6.1)

i

n The mode shape functions, φm w and φw were the beam bending mode shape functions

which were adapted based on the boundary conditions of the aluminum plate. The

153

CFCF boundary conditions were considered. In x direction, the mode shapes were the beam bending modes with clamped-clamped boundary conditions, shown in Eq. 4.29. Along the y direction, the beam modes with free-free boundary conditions were used: φn (y) = cosh(β n y) + cos(β n y) − λn (sinh(β n y) + sin(β n y)) λn =

cosh(β n c) − cos(β n c) sinh(β n c) − sin(β n c)

(6.2)

Here β n is determined using the characteristic equation for the free-free end boundary condition of a beam, which is: cos(β n c) cosh(β n c) = 1

(6.3)

The first 25 modes were included and m and n were mapped in Table 4.2. Based on the Kantorovich method, we have already solved the plate mode shape functions as shown in Table 5.5 and 5.6. There were only first seven modes needed in our calculation and these total mode shape numbers coincide with the mode numbers in experimental results. The frequency predictions using beam and plate modes were listed in Table 6.3 and were compared to experimental results. For the first mode, the errors were largest in both analyses, about 5%, and this is due to the boundary condition effects. From the second to seven modes, the frequency prediction errors decrease in both analyses and the error in the analysis using plate mode shapes were less than those in the analysis using beam modes except the seventh mode. Both analyses over-predicted the frequency, except the seventh mode which is expected. The experimental mode shape functions which were extracted using the Star software were presented for the first seven modes for 15 tested locations and Table 6.4 and 6.5 show the results. The mode shape functions are presented by the magnitude

154

and phase at each location because of damping exists in any real structures. The real components of the mode shapes were used in our analyses. We can reconstruct the mode shapes functions for the entire plate using two-dimensional interpolation based on information from these 15 points. The first six mode shape functions are plotted in contour form, as shown in Figures 6.5 a-f. From the figures, we can identify the nodal line and mode number clearly. The analytical mode shape functions predicted by the assumed modes method using plate modes are presented as well in Figures 6.6 a-f. We noted that the nodal lines in the experimental results were curved while the nodal lines predicted by the analyses are straight lines. We need more separable terms for each mode in our analyses to achieve the better prediction of plate bending vibration. Finally we compared the frequency response functions predicted by both analyses by picking up one location on the plate, the number 15, as shown in Figure 6.4. And the coordinates were listed in Table 6.1. The frequency response functions were plotted in Figure 6.7. Both analyses captured the trend of frequency response functions.

6.2.2

Plate with PCLD Treatment

As shown in Figure 6.3, the PCLD treatment was placed on the center of the plate, which fully covered the y direction and covered from x1 = 4in. to x2 = 8in. in x direction, breaking the whole plate into three regions. Region 1, (x, y) = [0 : 4; 0 : 10] is an isotropic aluminum plate; the PCLD region PCLD is three-layer sandwich plate; region 3, (x, y) = [8 : 12; 0 : 10], is an isotropic plate again. For the base plate, we assumed CFCF boundary conditions and the constraining plate had FFFF boundary conditions, in which FFFF denote a plate free along all edges. As with uniform

155

aluminum plate, the assumed modes method was used to calculate the response of a plate with PCLD treatment. In this case, we have to also include the in-plane mode shape functions as well. The assumed modes method was used to calculate the sandwich plate with PCLD treatment, in which the GHM method was adopted to account for the frequency dependent complex shear modulus of the viscoelastic core. Two analyses were developed, that is assumed modes using beam and rod modes, and assumed modes using plate modes for base plate and rod modes for constraining layer. 1. First we will demonstrate the assumed modes method using the beam and rod mode shapes, denoted as Analysis I. We assumed the transverse displacements were the same for each layer in the region of the PCLD treatment. Beam bending modes with clamped-clamped boundary conditions in x direction and beam bending modes with free-free boundary conditions were used to approximate the transverse displacement w(x, y), which were the same as the uniform aluminum plate case. Base plate had CFCF boundary conditions for in-plane motion. In-plane displacements, u1 (x, y) and v1 (x, y), were assumed to be: mπx nπy cos l c i  nπy mπx cos v1 (x, y) = V1i (t) sin l c

u1 (x, y) =



U1i (t) sin

(6.4)

i

which satisfied the geometric boundary conditions along the x and y directions. The constraining plate was a plate with FFFF configuration, and its in-plane displacements were mπx nπy cos l c i  nπy mπx cos v3 (x, y) = V3i (t) cos l c

u3 (x, y) =



U3i (t) cos

i

156

(6.5)

We also include the rigid body modes in our calculations. The mode number i , m and n are defined in Table 4.2. We can substitute these assumed modes and construct the mass, stiffness, and damping matrices as shown in the Appendix. The only difference is that we had to integrate by pieces to assemble the whole system matrices. The GHM method was used to account for the frequency dependent complex shear modulus of viscoelastic core. A total of 25 modes were included to achieve the frequency convergence compared to experimental results. This led to a model with 500 degrees of freedom because of additional internal coordinates used in the GHM method. 2. In order to save the computation cost, we try to improve the assumed modes by the plate bending and in-plane modes which were solved based on the Kantorovich method, which is denoted as Analysis II. These mode shapes minimize the total system energy and were higher order solutions. The transverse displacement w, were assumed as shown in the Table 5.5 and 5.6. In-plane displacements, u1 and v1 in the base plate, were assumed as the same in Table 5.7 to 5.10. These mode shape functions were solved based on the Kantorovich method for isotropic rectangular plates under bending and in-plane vibrations. For the constraining plate, we used same modes as given by Eq. 6.5, which are approximated by the rod modes in both x and y directions. We have tried to use the in-plane plate mode shape functions which were solved based on the Kantorovich method for the constraining plate. We found that more in-plane modes were needed compared to the case of using rod modes for approximation. Our goal is to alleviate the computational cost. Therefore, the combination of plate modes for

157

base plate and rod modes for the constraining plate was used in the assumed modes method for the sandwich plate analysis. Finally, for each displacements, the first 16 modes were included and led to a model with 320 degrees of freedom. We calculated the natural frequencies and loss factors and compared to the experimental data which was processed by the Star software [72]. These results were presented in Table 6.6 for frequency and Table 6.7 for loss factor. The mode shape functions calculated by the Star software were given in Table 6.8 and 6.9. The mode shapes plotted a contour form, are illustrated in Figure 6.8. The analytical mode shape functions were plotted in Figure 6.9 to compare with the experimental results. The frequency response functions predicted by Analysis I and Analysis II were plotted and compared against the experimental data at location 11, as shown in Figure 6.10. The analytical predictions captured the general trend of the frequency response functions for the plate with PCLD treatment and both magnitude and phase were correlated to the experimental results.

6.3

Summary

The analytical results of frequency, mode shapes and response have been validated by the experimental results. For the aluminum plate, the plate mode shape functions found based on the Kantorovich method improve our predictions. As shown in Table 6.3, there were only total seven modes needed to achieve the same accuracy as predicted by 25 beam bending mode shapes, because the plate mode shape functions

158

can minimize the total plate bending energy. The largest error appeared for the first mode in both methods and this is due to the boundary condition effects. We can minimize the error by improving the clamping design. From the second to the sixth modes, the error predicted by plate modes were less than those using the beam modes except the seventh mode. The experimental results for mode shapes were plotted in a contour form, as shown in Figure 6.5. We can identify the nodal line clearly and mode number can be determined for each mode. These mode shapes were generated from curve fitting results of 15 measured points, and we would need more points to obtain better results. The laser vibrometer will help us to scan the whole plate to obtain the response at many locations. For the given analytical results in Figure 6.6, we can see that the mode shapes demonstrated some symmetric properties. In order to improve these mode shape, we have to include more separable terms for each mode instead of a single term in our calculation of these mode shape functions. The frequency response function were plotted and compared to the experimental results for a given points. We select this point in order to demonstrate all the modes. Our predictions captured the trend and were shifted to right because of over-estimation of frequency. For the plate with PCLD treatment, we developed two analyses. For both analyses, the GHM method was incorporated to account for the frequency dependent complex shear modulus. The first was to use beam and rod modes for all the displacements in the assumed modes method, as shown in chapter 4. We include first 25 modes for each displacement and it leads to a system with about 500 degrees of freedom. The second was to use the plate modes for base plate structure and the rod modes were applied to in-plane modes for the constraining plate. We tried to use plate in-plane

159

modes with FFFF boundary for the constraining plate, but the results were not very promising. More in-plane modes for the constraining plate were needed in order to achieve comparable accuracy of frequency predictions compared to analysis I. Our goal is to reduce the number of modes included in the assumed mode method and maintain the accuracy of our predictions. Therefore, we developed the analysis II, in which we updated assumed modes for base plate using plate bending and in-plane modes which were solved based on Kantorovich method and used the rod modes to approximate the in-plane displacements in the constraining plate. In analysis II, only first 16 modes for each displacement are included and it leads to a system with about 320 degrees of freedom. As shown in Table 6.6, the frequency predictions for the first five modes in both analyses were compared to the experimental results. The largest error was in the first mode for both analyses, and both analyses achieved the same accuracy of frequency predictions. The total numbers of modes included in Analyses II were 80 while the number of modes were 125 for Analyses I. The natural frequency of PCLD plate decreased slightly compared to the frequency predictions in aluminum plate. However, we obtained higher damping for each modes, which can lower the vibrations in the structures. Table 6.7 shows the loss factors predicted by both analyses and are compared to the experimental data. The errors were large because we were dealing with very small number and those loss factors were associated with frequencies. The analytical damping predictions were lower than the experimental data because we can not include all the damping mechanism in our anlayses. In the Analyses II, the damping predictions were higher than those in the

160

Analyses I. The experimental mode shape functions for sandwich plate were plotted in Figure 6.8. The mode shapes functions were similar to those in Figure 6.5 for the uniform aluminum plate. This indicated that the bending mode shape would not change very much for these two cases. This validates our assumption to use isotropic bending plate mode shapes and the assumption that the whole sandwich plate exhibit same transverse displacement for each layers. The response functions predicted by both analyses captured the trend of experimental results for a point as shown in Figure 6.10. The presented analyses can predict the behaviors of the plate with PCLD treatment.

161

Table 6.3: Bending frequency results for an aluminum plate with CFCF boundary conditions Bending

Plate Modes i = 7

Beam Modes i = 25

Mode

Expt

Model

Error

Model

Error

No.

[Hz]

[Hz]

[%]

[Hz]

[%]

1,1

83.7

87.8

4.9

88.2

5.38

1,2

107.3

111.01

3.46

111.31

3.74

1,3

207.13 209.52

1.15

210.32

1.54

2,1

233.68 241.56

3.37

243.06

4.02

2,2

266.03 274.75

3.28

276.88

4.08

2,3

381.41

388.3

1.81

391.21

2.6

1,4

420.68 419.56

-0.27

420.85

0.04

162

Table 6.4: Experimental results of bending mode shape functions for an aluminum plate with CFCF boundary conditions, 15 tested locations from mode 1 to 4 Mode 1 Pts Mag.

Phase

Mode 2 Mag.

[deg.]

Mode 3

Phase

Mag.

[deg.]

Phase

Mode 4 Mag.

[deg.]

Phase [deg.]

1

1.44

-2.43

1.41

176.71

0.6696 -175.22

2.85

-179.35

2

1.41

0.213

0.675

176.11

0.0387

-13.57

1.76

178.93

3

1.13

-2.04

0.0103

166.13

0.4104

-2.28

1.54

178.74

4

1.16

-1.69

0.8706

1.09

0.068

12.63

1.72

177.63

5

1.24

-2.76

1.32

-2.34

0.4659

-179.5

2.25

176.9

6

2.6

-1.3

2.12

177.38

1.08

177.68

0.3122

0.0956

7

2.27

1

1.48

176.61

0.2334

51.6

8

2.29

-0.446 0.1307

145.15

0.6196 -0.8408 0.0261 -105.13

9

2.55

-0.997

1.35

-6.31

0.0769

-22.89

0.0231 -137.54

10

2.9

-0.277

2.85

-0.7736

1.13

177.4

0.0871

-3.35

11

2.58

-0.251

2.44

177.13

1.1

179.97

2.41

-3.82

12

1.87

-2.05

1.53

-176.32 0.0479

-1.97

1.48

-0.5379

13

2.18

-3.16

0.0557

-2.61

0.6643

-4.75

1.57

-2.83

14

2.11

-1.65

1.18

0.4232

0.162

-1.4

1.65

-0.4065

15

2.21

-1.9

2.53

-0.0681

1.04

171.22

1.82

-1.4

163

0.2943 -141.89

Table 6.5: Experimental results of bending mode shape functions for an aluminum plate with CFCF boundary conditions, 15 tested locations from mode 5 to 7 Mode 5 Pts

Mag.

Phase

Mode 6 Mag.

[deg.] 1

2.82

-0.9878

2

1.24

3.83

3

0.0558 -177.69

Phase

Mode 7 Mag.

[deg.] 1.63

-1.62

Phase [deg.]

1.7

0.0974 -173.77 0.6979

2.76 -173.9

1.1

-179.65

0.274

-3.13

1.05

1.23

4

1.17

178.64

0.1162

179.27

5

2.12

176.93

1.33

0.4478

0.8784 -172.69

6

0.3014 -175.96 0.2253 -170.85

2.73

1.48

7

0.1761

-68.55

0.2373 -118.33

1.97

-159.07

8

0.0415

160.76

0.0883

158.3

0.6531

168.06

9

0.1053

-58.9

0.0572

99.56

1.58

1.51

10

0.1918

0.8424

3.7

-178.9

11

1.96

178.74

1.46

178.71

2.72

4.96

12

0.9946

177.12

0.0093

-7.6

1.49

-175.69

13

0.0395

38.63

1

-1.79

0.1036

9.15

14

0.9903

-2.85

0.2673

-1.17

1.77

2.2

15

2

-0.8992

1.32

176.99

2.97

-176.13

0.2075 -173.33

164

Table 6.6: Bending frequency results for a plate with PCLD treatment, as shown in Figure 6.3; in analysis I, 25 modes for each displacement were assumed and it leads to 500 degrees of freedom; in analysis II, 16 modes for each displacement were used for a total of 320 degrees of freedom. Bending

Analysis I

Analysis II

Mode

Expt

Model

Error

Model

Error

No.

[Hz]

[Hz]

[%]

[Hz]

[%]

1,1

83.1

87.7

5.54

87.8

5.66

1,2

104.91 107.92

2.87

108.02

2.96

1,3

218.9

220.82

0.88

220.74

0.84

2,1

234.2

241.9

3.29

241.17

2.98

2,2

277.8

286.28

3.06

285.59

2.81

165

Table 6.7: Loss factor results for a plate with PCLD treatment, as shown in Figure 6.3; in analysis I, 25 modes for each displacement were assumed and it leads to 500 degrees of freedom; in analysis II, 16 modes for each displacement were used for a total of 320 degrees of freedom. Mode

Analysis I

Analysis II

No.

Expt.

Model

Error [%]

Model

Error [%]

1,1

0.06

0.0448

-25.3

0.0486

-19

1,2

0.0412 0.0334

-18.9

0.0371

-9.96

1,3

0.0424 0.0275

-35.1

0.0304

-28.1

2,1

0.031

0.0302

-2.58

0.0328

5.81

2,2

0.0334 0.0297

-11.1

0.039

16.8

166

Table 6.8: Experimental results of bending mode shape functions for a plate with PCLD treatment; 15 tested locations from mode 1 to 3 Mode 1 Pts

Mag.

Mode 2

Phase

Mag.

[deg.]

Phase

Mode 3 Mag.

[deg.] 1.56

Phase [deg.]

1

1.1

8.49

-173.99 0.6049

2.54

2

0.9012

2

3

0.8187

2.17

0.1127

1.85

0.3855 -125.34

4

0.8355

0.4081

1.23

-5.41

0.0503

-52.01

5

0.8747

3.72

2.32

0.0929

0.7609

-16.84

6

1.96

2.95

2.77

177.71

0.5755

18.03

7

1.59

-2.64

1.3

178.41

0.0777 -137.47

8

1.32

-1.66

0.1907

-10.01

0.358

-176.83

9

1.23

-2.26

1.33

-0.5522

0.119

-171.19

10

1.28

-0.5705

3.52

0.9492

0.6885 -0.3351

11

1.72

-1.89

2.43

179.51

0.376

18.11

12

1.39

-1.2

1.31

178.56

0.08

128.38

13

1.25

-17.39

0.4547

174.87

14

0.847

-2.06

1.51

2.35

0.0494

126.74

15

1.02

-2.77

2.97

0.832

0.6653

26.98

0.7458 -168.24 0.0904 -118.61

-0.0357 0.0883

167

Table 6.9: Experimental results of bending mode shape functions for the plate with PCLD treatment; 15 tested locations from mode 4 to 5 Mode 4 Pts

Mag.

Mode 5

Phase

Mag.

[deg.]

Phase [deg.]

1

2.27

-168.49

2.82

3.13

2

1.74

-172.04

1.65

11.6

3

1.54

-172.26 0.1087

37.86

4

1.5

-179.72

2.03

177.38

5

1.93

-175.29

3.51

179.9

6

0.0397

-56.12

0.2074

-38.21

7

0.2324

167.67

0.2046

-28.23

8

0.1221

164.13

0.0073

-83.08

9

0.365

165.57

0.3651

178.4

10

0.1337 -105.82 0.1462

170.5

11

1.38

-3.18

2.06

-178.29

12

1.59

-2.84

1.58

179.63

13

1.05

8.96

0.0783

-42.18

14

1.26

-1.56

1.58

-2.44

15

1.29

3.16

2.67

2.59

168

10

10

9

9

8

8

7

7

.75

−0

.5

−0.2

5

6

0.25

0.5

0.75

0.75

0.5

4

4

3

3

2

2

1

1

0 0

0 0

2

4

6

8

10

12

a) ω = 83.7Hz; mode (1, 1) 10

0.5 0.75 2

4

6

8

10

12

b) ω = 107.30Hz; mode (1, 2) 0.75

−0 .75

9

−0.25

8

0.25

10

−0.7 5 −0.5

9

0

0

5

8

0

0

7

7

0.25

0.5

0.25

0

0.5

10

2

5

1 0 0

8

3

0

0

2

−0.25

4

0.25

3

−0.5

5

0.5 4

0.25

6

5

−0.25

6

−0.5

5

0.25

6

−0

2

−0.2

−0.5

4

6

1

−0.75 8

10

12

0 0

2

4

6

12

c) ω = 207.18Hz; mode (1, 3) d) ω = 233.68Hz; mode (2, 1) 10

10

0

25

0.25

0.5

−0.5 −0.25 0

0

8

0 7

0.25 6

0 0

5

5

−0.25

0

0

0.

9

−0.5

75

7 6

−0.75

0.

8

0.5

5

5

0.7

−0.2

9

0.5

−0

.5

4

4

5 0.2

3

3

0

2

2

−0.5

1 0 0

0 0.25

0.5

0.5

1 2

4

6

8

10

12

e) ω = 266.03Hz; mode (2, 2)

0 0

0

−0.25

2

4

6

0 −0.25 −0.5 8

10

12

f) ω = 381.40Hz; mode (2, 3)

Figure 6.5: Contour plot of experimental bending mode shape functions for an aluminum plate with CFCF boundary conditions

169

9

8

8

7

7

6

6

−0.5

−0.5

−0.75

4

−0.25

5

−0.25

10

9

−0.75

10

3 2

1

1 4

6

8

10

0

0

4

2

2

0.5 0.25

5

3

0 0

0.75

12

a) ω = 87.8Hz; mode (1, 1) 10

−0.25

−0.5

−0.75

0 0

2

4

6

8

10

12

b) ω = 111.01Hz; mode (1, 2) 10

0.75 0.5

9

0

0.25

1

−0.25

−0.5

−0.75

−0.25

3

−0.25

2

−0.75

3

0.75

4

−0.5

0.25

5

4

0.25

6

5

−0.5

7

6

0.5

7

0

8

0

0.75

8

0.25

0.5

9

2

0.5

1

0.75 0 0

2

4

6

8

10

12

0 0

2

4

6

8

10

12

c) ω = 209.52Hz; mode (1, 3) d) ω = 241.56Hz; mode (2, 1) 10

10

−0.7

0.75 8

3

−0.5

0.5

−0.75

0.75

0 0

2

4

6

8

10

12

e) ω = 274.75Hz; mode (2, 2)

2

0

1

0.5

0 0

75

0.

5

.5

−0

25

5 0.2

0.

−0

4

2 1

.75

5

−0

5

2 −0.

0

6

0

0

3

7

0.

0

5

0

0

25

−0.

0.25

6

4

−0.5

0

0.5

−0.5

−0.25

0.25

8

7

5

0.5

9

−0.75

0.75

.2 5

9

0 −0.25

0.2

5

−0.5 5 −0.7

.75 2

4

6

8

10

12

f) ω = 388.3Hz; mode (2, 3)

Figure 6.6: Contour plot of analytical bending mode shape functions for an aluminum plate with CFCF boundary conditions

170

Legend: − Expt. −− Plate Modes −. Beam Modes

Mag(M/N) [log10]

−3 −4 −5 −6 −7

100

150

200

100

150

200

250

300

350

400

450

250 300 Frequency [Hz]

350

400

450

Phase [deg]

200 150 100 50 0

Figure 6.7: Frequency response functions of an aluminum plate with CFCF boundary conditions, at location 15, as shown as Table 6.1; in which only 7 plate modes were included and 25 beam bending modes were used

171

10

10

9

9

−0.75

−0.5 8

5

0.7

8

7

6

6

0.25

0.5

7

5

−0.25

0 0

5

0. 5

0.25

4

4

0.25 3

3

2

2

1

1

0 0

2

4

6

8

10

10

0 0

12

a) ω = 83.1Hz; mode (1, 1)

2

8

10

12

−0.7

9

0.25

5

8

0

0

7

7

5

−0.2

5

0.

3

0.25 0.5

−0.5

4

0

3

−0.25

−0.5

−0.25

5

.5

0.25

6

−0

5

0

6

4

6

b) ω = 104.91Hz; mode (1, 2)

0.5

8

4

10

0.75

9

0.5 0.75

0 2

0.25

0 0

1

0.75 2

4

6

8

10

0 0

12

c) ω = 218.9Hz; mode (1, 3) 10

.7

5

0.5

1

−0

2

2

4

6

8

10

12

d) ω = 234.2Hz; mode (2, 1)

.5

0.75

0.25

8

0

0.5

−0

9

7

−0.25 6

0

0

4

.25

2

−0 .5

1

−0

0 0

0

−0

3

0.25

5

5

.7

2

4

6

0.5

8

10

12

e) ω = 277.8Hz; mode (2, 2) Figure 6.8: Contour plot of experimental bending mode shape functions for a plate with PCLD treatment under CFCF boundary conditions

172

10

10

9

9

8

8

7

7

−0.75 −0.5 −0.25

−0.5

−0.75

6

−0.25

−0.5

−0.75

5

−0.25

6

0

5

4

4

3

3

2

2

1

1

0.25

0 0

2

4

6

8

10

0.5

0.75

0 0

12

a) ω = 87.8Hz; mode (1, 1)

2

4

6

8

10

12

b) ω = 108.02Hz; mode (1, 2)

10

10

9

−0.5

−0

.25

8

−0.75

9 8

−0.5

0.5

0.5

0.75

−0.5

−0.75

4

0.25

3

0

5

0. 75 0.5 0.2 5

4

−0.25

6

5

0.25

7

6

−0.25

0 7

3

0 2

−0.5

1

2

−0.2

5

1

−0.75

0 0

2

4

6

8

10

0 0

12

2

4

6

8

10

12

c) ω = 220.74Hz; mode (1, 3) d) ω = 241.17Hz; mode (2, 1) 10

5

−0.75

8

0.5

7 6

0.7

9

0

−0.5

−0

.25

0.25 0

5

0

0.25

4 3

−0. −0.5

.75

0.75

2

25

0

0.5

−0

1 0 0

2

4

6

8

10

12

e) ω = 285.59Hz; mode (2, 2) Figure 6.9: Contour plot of analytical bending mode shape functions for a plate with PCLD treatment under CFCF boundary conditions

173

Legend: − Expt. −− Analysis II −. Analysis I

Mag(M/N) [log10]

−3.5 −4 −4.5 −5 −5.5 −6

100

150

200

250

300

100

150 200 Frequency [Hz]

250

300

Phase [deg]

200 150 100 50 0

Figure 6.10: Frequency response functions of a plate with PCLD, at location 11; in analysis I, 25 modes for each displacement were assumed and it leads to 500 degrees of freedom; in analysis II, 16 modes for each displacement were used for a total of 320 degrees of freedom.

174

Chapter 7

Summary and Conclusions

The objectives of this research focused on the solution of vibration problems in sandwich beams and plates, and to validate all the analytical results with experimental data. To this end, the original contributions of this study are: • development of a spectral finite element method for the sandwich beam analyses; • analytical validation of the spectral finite element model using results from the assumed modes method and conventional finite element method for sandwich beam; • experimental validation for all the analyses of sandwich beams by comparing the results of natural frequency and response; • introduction of the GHM method in the assumed modes method for sandwich plate analyses using beam and rod modes; • achievement of plate bending and in-plane vibration mode shape functions for isotropic rectangular plate based on the Kantorovich method;

175

• application of plate mode shape functions to update the assumed modes in the sandwich plate analyses; • experimental validation of sandwich plate analyses using the results of natural frequency, loss factor, mode shape function, and response. Thus the goals of our research have been achieved.

7.1

Sandwich Beam

A spectral finite element method (SFEM) was developed for the sandwich beam analyses. The frequency dependent complex shear modulus of the viscoelastic core was implicitly accounted because the SFEM was developed in frequency domain. There is no need for additional damping model. The shape functions in the SFEM were duplicated from the progressive wave solutions. Therefore, the number of elements needed in SFEM coincides with the number of different impedance in the structures. The conventional finite element method (CFEM) and the assumed mode method (AM) were used to calculated the sandwich beams as well where the GHM method has to be included to account for the frequency dependent complex shear modulus of viscoelastic core. This leads to a large size of model because the additional internal coordinates in the GHM method increase the degrees of freedom in the analyses of sandwich beam The analytical results of natural frequency and frequency response were validated by the experimental data by testing two beam specimens with 50% and 75% PCLD treatment. The SFEM can provide an accurate solution at the less computation cost compared to CFEM and AM.

176

7.2

Sandwich Plate

We expect to extend the SFEM for sandwich plate analyses. But it is extremely to obtain the exact solutions of equations for sandwich plate. The assumed modes method was developed to model the sandwich plate and the GHM method was incorporated where one-dimensional beam and rod modes were approximated the two-dimensional plate modes in both x and y directions. This approach consumes a large computational cost. In order to improve this simple approach, we updated the assumed modes using plate modes which were solved from isotropic rectangular plate based on the Kantorovich method. These updated plate bending and in-plane mode shape functions were a higher order approximation of the biharmonic equation in plate bending and Navier equations in plate in-plane vibration. The final solutions of these plate modes were presented and the approach was demonstrated for plate bending and in-plane vibrations. We introduced these plate modes in the assumed modes method for the sandwich plate analyses. The number of modes included decreases compared to the case of using beam and rod modes. Experiments were conducted to study the sandwich plate dynamics. The experimental data of natural frequency, loss factor, mode shape functions and response were presented. These results were used to validated our analytical predictions. Good correlations were achieved between the analyses and experiment. Therefore, the updated assumed modes method using plate modes can be used to analyze the sandwich plate structures.

177

7.3

Recommendations for Future Research

This study has demonstrated that our approaches for the sandwich beam and plate analyses were successful. All the results were experimental validated. The SFEM in the sandwich beam analyses can provide exact solutions for the corresponding governing equations and we can implicitly account for the frequency dependent complex shear modulus of the viscoelastic core. We expect to extend the spectral finite element approach to sandwich plate analyses as well. Our next step will focus on directly solving to the PDEs of sandwich plates in order to apply the SFEM method to sandwich plate analysis. The Kantorovich method has been applied to the problems of isotropic rectangular plates under bending and in-plane vibrations. We expect to extend this approach to sandwich plate analysis as well. This will provide us the coupled mode shape functions for all the transverse and in-plane displacements. However, the advanced computational schemes are needed in order to solve these fully coupled PDEs with the complex coefficients introduced by the complex shear modulus of the viscoelastic core. On the other hand, we want to improve plate mode shape functions of isotropic plate structures under bending and in-plane vibrations. Now we assume a single separable variable solution form. We could include more terms to improve current results, especially, for plate in-plane vibration with free edges. The essentials of wave propagation in the plate structures are needed to be well studied for both isotropic plate and sandwich plate structures. Based on these wave solution forms, a new type of finite element approach will be produced for dynamic analyses of two-dimensional plate structures. In all our studies, we assume that a structure has a uniform cross section area.

178

For non-uniform beams, the wave solutions will be special mathematical functions, such as Bessel functions. However, for non-uniform plates, it is very difficult to solve it analytically. And it is still a challenge to solve it using the Kantorovich method. Therefore, the next step is to study the non-uniform structures using spectral finite element method or the Kantorovich method. The motivation of this research is developed a hybrid noise control scheme. Therefore, based on our approaches, we can develop a comprehensive acoustic and structural coupled system to study the vibration and noise control. The control algorithm can be design to achieve the “jet smooth quiet ride” goal in the helicopter industry.

179

Appendix A

Mass and Stiffness Matrices

The elements of the mass and stiffness matrices for a sandwich plate with isotropic faces and a viscoelastic core are listed here. The mass and stiffness matrices for the sandwich plate may be constructed in blocks or sub-matrices using the total energy, and assumed mode shape functions. We assumed that the five displacements for sandwich plate motion were expansion of associated mode shapes. These mode shapes are either adapted from beam and rod mode shapes or plate mode shapes which were solved from uniform isotropic plate bending and in-plane motions based on the Kantorovich method. The displacements are: w(x, y, t) =



Wi (t)Φiw (x, y)

i

u1 (x, y, t) =



Ui1 (t)Ψiu1 (x, y)

i

v1 (x, y, t) =



V1i (t)Ψiv1 (x, y)

i

u3 (x, y, t) =



U3i (t)Ψiu3 (x, y)

i

v3 (x, y, t) =

 i

180

V3i (t)Ψiv3 (x, y)

We substitute the above displacements into the total energy expression of sandwich plate, Eqs: 4.5 and 4.7. The final discretized equations of motions can be expressed as: ¨ + Ke x + G⋆ Kv x = F Mx

(A.1)

where M is the mass matrix, Ke and Kv are the stiffness matrices which are contributed from elastic and viscoelastic part, respectively. x is modal coefficients vector, in which the components correspond to the amplitudes of assumed mode shapes. F is a force vector which is discretized by the assumed mode shape functions. The details of mass and stiffness matrices are demonstrated in the following sections. When we construct the Eq. A.1, the GHM method has to be introduced to account for the frequency dependent complex shear modulus, G⋆ . This has already been illustrated in Section 2.3.3.

A.1

Mass Matrix

The mass matrix is shown as: 

 Mww Mwu1   . .. M = .  ..   Mv3 w Mv3 u1

181

··· ..

.

···



Mwv3    ..  .    Mv3 v3

(A.2)

The off diagonal sub-matrices are all zero. Mwu1 , Mwv1 , Mwu3 , Mwv3 = 0 Mu1 w , Mu3 w , Mv1 w , Mv3 w = 0 Mu1 v1 , Mu1 u3 , Mu1 v3 , Mv1 u1 = 0 Mu3 u1 , Mv3 u1 , Mv1 u3 , Mv1 v3 = 0 Mu3 v1 , Mv3 v1 , Mu3 v3 , Mv3 u3 = 0

(A.3)

and the diagonal blocks are Mww (i, j) =




ρhΦiw Φjw dA

(A.4)




ρ1 h1 Ψiu1 Ψju1 dA

(A.5)




ρ1 h1 Ψiv1 Ψjv1 dA

(A.6)




ρ3 h3 Ψiu3 Ψju3 dA

(A.7)




ρ3 h3 Ψiv3 Ψjv3 dA

(A.8)

A

Mu1 u1 (i, j) =

A

Mv1 v1 (i, j) =

A

Mu3 u3 (i, j) =

A

Mv3 v3 (i, j) =

A

182

A.2

Stiffness Matrices

Before we show the formulas for the element of stiffness matrices, we first define some parameters. They are: Dt = g1 = g3 = A1 = A3 = d = ′ = ∗ =

E3 h33 E1 h31 + 12(1 − ν 2 ) 12(1 − ν 2 ) E1 2(1 + ν) E3 2(1 + ν) E1 (1 − ν 2 ) E3 (1 − ν 2 ) h1 + h3 + 2h2 2h2 ∂ ∂x ∂ ∂y

(A.9)

The stiffness matrices, Ke and Kv are assembled as some block matrices:    Kww,e Kwu1 ,e · · ·   .. .. .. Ke =  . . .    Kv3 w,e Kv3 u1 ,e · · · 

 Kww,v Kwu1 ,v · · ·   .. .. .. Kv =  . . .    Kv3 w,v Kv3 u1 ,v · · ·

183

Kwv3 ,e    ..  .    Kv3 v3 ,e

(A.10)



Kwv3 ,v    ..  .    Kv3 v3 ,v

(A.11)

and the first row of stiffness sub-matrices in Ke are: Kww,e(i, j) =




′′

′′

′′

Dt Φiw Φjw + 2νΦiw Φjw

∗∗

∗∗

+ Φiw Φjw

∗∗

A

+

 g1 h31 + g3 h33 i ′∗ i ′∗ Φw Φw dA 3

! (A.12)

Kwu1 ,e , Kwv1 ,e , Kwu3 ,e , Kwv3 ,e = 0

(A.13)

The second row of stiffness sub-matrices in Ke are: Ku1 u1 ,e (i, j) =




A1 Ψiu1 Ψju1 + g1 h1 Ψiu1 Ψju1




# ′ ∗ ∗ ′ A1 νΨiu1 Ψjv1 + g1 h1 Ψiu1 Ψjv1 dA

′

′

∗

∗

A

Ku1 v1 ,e (i, j) =

A

#

dA

Ku1 w,e , Ku1 u3 ,e , Ku1 v3 = 0

(A.14) (A.15) (A.16)

The third row of stiffness sub-matrices in Ke are: Kv1 u1 ,e = Ku1 v1 ,e T
 # ∗ ∗ ′ ′ A1 Ψiv1 Ψjv1 + g1 h1 Ψiv1 Ψjv1 dA Kv1 v1 ,e (i, j) =

(A.17)

Kv1 w,e , Kv1 u3 ,e , Kv1 v3 ,e = 0

(A.19)

(A.18)

A

The fourth row of stiffness sub-matrices in Ke are: Ku3 u3 (i, j) =




A3 Ψiu3 Ψju3 + g3 h3 Ψiu3 Ψju3




# ′ ∗ ∗ ′ A3 νΨiu3 Ψjv3 + g3 h3 Ψiu3 Ψjv3 dA

′

′

A

Ku3 v3 ,e (i, j) =

A

∗

∗

#

dA

Ku3w,e , Ku3 u1 ,e , Ku3 v1 ,e = 0

(A.20) (A.21) (A.22)

The fifth row of stiffness sub-matrices in Ke Kv3 u3 ,e = Ku3 v3 ,e T
 # ∗ ∗ ′ ′ A3 Ψiv3 Ψjv3 + g3 h3 Ψiv3 Ψjv3 dA Kv3 v3 ,e (i, j) = A

184

(A.23) (A.24)

Kv3 w,e , Kv3 u1 ,e , Kv3 v1 ,e = 0

(A.25)

The first row of stiffness sub-matrices in Kv are: Kww,v (i, j) =




A

Kwu1 ,v (i, j) = −

#  ′ ′ ∗ ∗ d2 h2 Φiw Φjw + Φiw Φjw dA


dΦiw Ψju1 dA




dΦiw Ψjv1 dA

(A.26)

′

(A.27)

∗

(A.28)

A

Kwv1 ,v (i, j) = −

A

Kwu3 ,v (i, j) =




dΦiw Ψju3 dA




dΦiw Ψjv3 dA

′

(A.29)

∗

(A.30)

A

Kwv3 ,v (i, j) =

A

The second row of stiffness sub-matrices in Kv are: Ku1 w,v = Kwu1 ,v T
1 i j Ψ Ψ dA Ku1 u1 ,v (i, j) = h2 u1 u1

(A.31) (A.32)

A

Ku1 v1 ,v = 0 Ku1 u3 ,v (i, j) = −

(A.33)


A

1 i j Ψ Ψ dA h2 u1 u3

Ku1 v3 ,v = 0

(A.34) (A.35)

The third row of stiffness sub-matrices in Kv are: Kv1 w,v = Kwv1 ,v T Kv1 u1 ,v = Ku1 v1 ,v T
1 i j Ψ Ψ dA Kv1 v1 ,v (i, j) = h2 v1 v1

(A.36) (A.37) (A.38)

A

Kv1 u3 ,v = 0 Kv1 v3 ,v (i, j) = −

(A.39)


A

185

1 i j Ψ Ψ dA h2 v1 v3

(A.40)

The fourth row of stiffness sub-matrices in Kv are: Ku3 w,v = Kwu3 ,v T

(A.41)

Ku3 u1 ,v = Ku1 u3 ,v T

(A.42)

Ku3 v1 ,v = 0
1 i j Ψ Ψ dA Ku3 u3 ,v (i, j) = h2 u3 u3

(A.43) (A.44)

A

Ku3 v3 ,v = 0

(A.45)

The fifth row of stiffness sub-matrices in Kv are: Kv3 w,v = Kwv3 ,v T

(A.46)

Kv3 u1 ,v = 0

(A.47)

Kv3 v1 ,v = Kv1 v3 ,v T

(A.48)

Kv3 u3 ,v = 0
1 i j Kv3 v3 ,v (i, j) = Ψ Ψ dA h2 v3 v3

(A.49) (A.50)

A

All the elements in the mass and stiffness matrices has been demonstrated. Finally, the GHM method was used to account for the frequency dependent complex shear modulus. We have shown this in Section 2.3.3

186

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